Diagonal checker-jumping and Eulerian numbers for colorsigned permutations, Electron

合集下载

cheat

3 Let c = 2 . Then we have m−1 i=0
n
c =n
i
= 2n(c(k−1) logc n − 1) = 2nk − 2n,
k
= 2n(clog2 n − 1)
c −1 c−1
m
xi .
We choose G(x) = i≥0 xi gi . Rewrite in terms of G(x): G(x) − g0 = 2G(x) + xi . x
k
Trees 39. 41. 43. x = x−n
k n
k
n k n k =
k=0
k m
n
=
k=0
k n−k = n! n m
n
k=0
1 k , k! m
k=0
n k
x+k , 2n
k+1 (−1)n−k , m+1
m
n = m
n+1 k+1
m
k (−1)m−k , m n+k , k
Every tree with n vertices has n − 1 edges. Kraft inequality: If the depths of the leaves of a binary tree are d1 , . . . , dn :
i=
i=1 n
n(n + 1) , 2
n
i2 =
i=1
n(n + 1)(2n + 1) , 6
n
n
i3 =
i=1
n2 (n + 1)2 . 4
In general: im =
i=1 n−1 i=1

图论及其应用

图和子图图和简单图图 G = (V, E)，其中 V = {νv v v ,......,,21} V ---顶点集， ν---顶点数E = {e e e 12,,......,ε}E ---边集， ε---边数例。

左图中， V={a, b,......,f}, E={p,q, ae, af,......,ce, cf} 注意，左图仅仅是图G 的几何实现（代表），它们有无穷多个。

真正的图G 是上面所给出式子，它与顶点的位置、边的形状等无关。

不过今后对两者将经常不加以区别。

称边 ad 与顶点 a (及d) 相关联。

也称顶点 b(及 f) 与边 bf 相关联。

称顶点a 与e 相邻。

称有公共端点的一些边彼此相邻，例如p 与af 。

环（loop ，selfloop ）：如边 l 。

棱（link ）：如边ae 。

重边：如边p 及边q 。

简单图：（simple graph ）无环，无重边平凡图：仅有一个顶点的图（可有多条环）。

一条边的端点：它的两个顶点。

记号：νε()(),()().G V G G E G ==。

习题1.1.1 若G 为简单图，则εν≤⎛⎝ ⎫⎭⎪2 。

1.1.2 n ( ≥ 4 )个人中，若每4人中一定有一人认识其他3人，则一定有一人认识其他n-1人。

同构在下图中，图G 恒等于图H , 记为 G = H ⇔ V （G)=V(H), E(G)=E(H)。

图G 同构于图F ⇔ V(G)与V(F), E(G)与E(F)之间各存在一一对应关系，且这二对应关系保持关联关系。

记为 G ≅F 。

注往往将同构慨念引伸到非标号图中，以表达两个图在结构上是否相同。

de f G = (V, E)y z w cG =(V , E )w cyz H =(V ’, E ’)’a ’c ’y ’e ’z ’F =(V ’’, E ’’)注判定两个图是否同构是NP-hard 问题。

完全图(complete graph) Kn空图（empty g.） ⇔ E = ∅ 。

zoj poj ural分类怕下次找麻烦就分享了

zoj poj ural分类怕下次找麻烦就分享了ural, pku, zju -- 题目分类感谢mugu 的提供.... Ural Problem Set Volume 1: 1000-1099题号标题难度系数算法1000 A+B Problem 10% 直接加1002 Phone Numbers 50% 动态规划或最短路1003 Parity 70% 区间减法1004 Sightseeing trip 60% 最短路1005 Stone Pile 30% 动态规划或搜索1006 Square Frames 35% 模拟1007 Code Words 30% 模拟1008 Image encoding 30% 广度优先搜索1009 K-Based Numbers 20% 递推或枚举（数据规模小）1010 Discrete Function 40% 贪心1011 Conductors 25% 搜索1012K-Based Numbers 30% 递推1013 K-Based Numbers 33% 递推1014 The Product of Digits 30% 贪心1015 Test the differences 35% 模拟1016 A cube on the walk 50% 搜索或者最短路1017 The Staircases 30% 递推（母函数）1018 The Binary Apple Tree 50% 动态规划1019 A Line painting 40% 离散化处理1020 Rope 45% 一般的计算几何问题1021 Sacrament of the sum 40% 动态规划1022 Genealogical Tree 30% 拓补排序1023 Buttons 50% 动态规划1024 Permutations 25% 置换（和题目名字一样）1025 Democracy in Danger 20% 贪心1026 Questions and Answers 40% 快速排序1027 D++ again 40% 字符串处理1028 Stars 75% 线段树1029 Ministry 55% 动态规划（注意优化）1030 Titanic 60% 计算几何（注意精度！）1031 Railway Tickets 45% 动态规划1032 Find a mutilple 55% 同余问题1033 Labyrinth 30% 搜索、遍历1034 Queens in peaceful positions 35% 搜索1035 Cross-sitich 60% 欧拉路径问题1036 Lucky tickets 50% 递推1037 Memory Management 55% 模拟（注意优化）1038 Spell Checker 45% 字符串的操作及统计1039 Anniversary party 40% 树的动态规划1040 Airline company 55% 搜索1041 Nikifor 85% 线性代数问题（过了，但怀疑算法不对）1042 Central Heating 50% 解线性同余方程组1043 Cover an Arc. 58% 计算几何1044 Lucky tickets 40% 递推1045A funny game 55% 搜索1046 Geometrical dreams 88% 很复杂的计算几何1047Simple calculations 30% 数学题，解方程吧！1048 Superlong Sums 48% 高精度加法1049 Brave ballonists 30% 模拟1050 Preparing an article 63% 比较麻烦的字符串处理1051 A simple game on a grid 60% 数学杂题1052 Rabbit Hunt 28% 简单的计算几何1053 Pinocchio 32% 求最大公约数（题目意思不明）1054 Hanoi Tower 50% 数学问题1055 Combinations 40% 数学问题1056 Computer net 55% 动态规划1057 Amount of dergess 53% 数学问题1058 Chocolate 85% 复杂的计算几何问题1059 Expression 20% 简单的打印问题1060 Flip game 35% 我搜索，过了1061 Buffer Manager 45% 模拟1062 Triathlon 83% 复杂的不等式问题1064 Binary Search 48% 搜索1065 Frontier 75% 动态规划1066 Garland 40% 数学问题1067 Disk Tree 45% 模拟或排序1068 Sum 20% 简单的加法题1069 The prufer code 50% 构造1070 A local time 60% 考虑要周全1071 Nikifor - 2 45% 搜索1072 Routing 40% 最短路问题1073 A square problem 50% 动态规划1074 A very short problem 58% 模拟、判断1075 A thread in the space 65% 麻烦的计算几何1076 Trash 50% 二分图的最大权匹配1077 Travelling Tours 60% 图的遍历1078 Segments 40% 动态规划1079 Maxium 50% 数学问题1080 Map coloring 35% 深度优先搜索1081 Binary Lexicographic Sequence 40% 数学问题1082 Gaby Ivanushka 15% 没有算法，直接输出答案1083 Factorials!!! 25% 简单的数学计算1084 A goat in a kitchen garden 35% 简单的计算几何1085 Meeting 55% 最短路1086 Cryptography 45% 数学问题1087 The time to take stones 48% 递推1088 Ilya Murumetz 33% 二叉树的性质1089 verification with a vocabulary 50% 字符串处理1090 In the army now 60% 平衡二叉树1091 Tmutarakan exams 57% 运用容斥原理1092 Transversal 75% 贪心1093 Darts 65% 立体解析几何1094 E-screen 40% 字符串处理1095 Nikifor - 3 45% 同余问题1096 Get the right route plate! 40% 广度优先搜索1097 Square country - 2 63% 离散化处理1098 Questions 48% 字符串处理1099 Work scheduling 60% 图的最大基数匹配详情请见：acm - Ural专页 Posted: 2007-02-16 17:39 | [楼主] 威士忌我本楚狂人－凤歌笑孔丘级别: 管理员精华: 1 发帖: 1529 威望: 1652 点金钱: 80262 HDOJ币贡献值: 3 点在线时间:827(小时) 注册时间:2006-03-21 最后登录:2008-10-14poj--题目分类1、排序1423, 1694, 1723, 1727, 1763, 1788, 1828, 1838, 1840, 2201, 2376, 2377, 2380, 13 18, 1877, 1928, 1971, 1974, 1990, 2001, 2002, 2092, 2379, 1002（需要字符处理，排序用快排即可）1007（稳定的排序）2159（题意较难懂）2231 2371（简单排序）2388（顺序统计算法）2418（二*排序树）2、搜索、回溯、遍历1022 1111 1118 1129 1190 1562 1564 1573 1655 2184 2225 2243 2312 2362 2378 23861010,1011,1018,1020,1054,1062,1256,1321,1363,1501，1650,1659,1664,1753,2078,208 3,2303,2310,2329 简单：1128, 1166, 1176, 1231, 1256, 1270, 1321, 1543, 1606, 1664, 1731, 1742, 17 45, 1847, 1915, 1950, 2038, 2157, 2182, 2183, 2381, 2386, 2426, 不易：1024, 1054, 1117, 1167, 1708, 1746, 1775, 1878, 1903, 1966, 2046, 2197, 23 49, 推荐：1011, 1190, 1191, 1416, 1579, 1632, 1639, 1659, 1680, 1683, 1691, 1709, 17 14, 1753, 1771, 1826, 1855, 1856, 1890, 1924, 1935, 1948, 1979, 1980, 2170, 2288 , 2331, 2339, 2340,1979（和迷宫类似）1980（对剪枝要求较高）3、历法1008 2080 （这种题要小心）4、枚举1012，1046，1387，1411，2245，2326，2363，2381，1054（剪枝要求较高），1650 （小数的精度问题）5、数据结构的典型算法容易：1182, 1656, 2021, 2023, 2051, 2153, 2227, 2236, 2247, 2352, 2395, 不易：1145, 1177, 1195, 1227, 1661, 1834, 推荐：1330, 1338, 1451, 1470, 1634, 1689, 1693, 1703, 1724, 1988, 2004, 2010, 21 19, 2274, 1125(弗洛伊德算法) ，2421（图的最小生成树）6、动态规划1037 A decorative fence、1050 To the Max、1088 滑雪、1125 Stockbroker Grapevine、1141 Brackets Sequence、1159 Palindrome、1160 Post Office、1163 The Triangle、1458 Common Subsequence、1579 Function Run Fun、1887 Testing the CATCHER、1953 World Cup Noise、2386 Lake Counting 7、贪心1042, 1065, 1230, 1784,1328 1755（或用单纯形方法），2054，101 7，1328，1862，1922 ，2054，2209，2313，2325，2370。

数值分析英文课件

ˆ ∆y = y − y = 1.4 − 1.41421L ≈ 0.0142
or relative forward error of about 1 percent. Since 1.96 = 1.4 , the absolute backward error is
ˆ ∆x = x − x = 1.96 − 2 = 0.04
Computational error = Truncation error + rounding error
• Propagated （传播） vs. computational error 传播）
– x = exact value, – f = exact function,
ˆ x = approx. value ˆ f = its approximation
Backward vs. forward errors
Suppose we want to compute y = f ( x ) , where f : ℜ → ℜ ˆ but obtain approximate value y
Forward Error:
ˆ ˆ ∆y = y − y = f ( x ) − f ( x )
Example of Ill-Posed Problem
x 1 x 1 x 11 1 + 2 + 3 = 2 3 6 1 1 1 13 x1 + x2 + x3 = 3 4 12 2 1 x1 + 1 x2 + 1 x3 = 47 3 4 5 60
2 significant digits rounding
• Problems that are not well-posed are ill-posed.

信息学奥赛试题精选33题(附带题解)

第1~10题为基础题，第11~20题为提高题，第21~33为综合题注：因为在本文档中需要用到一些特殊的数学符号（如：求和号、分数等），所以当您在百度文库中浏览时，一些数学符号可能会显示不出来，不过当您把本文档下载下来在本地浏览时，所有的符号即可全部都显示出来。

^_^基础题：【1 Prime Frequency】【问题描述】给出一个仅包含字母和数字（0-9, A-Z 以及a-z）的字符串，请您计算频率（字符出现的次数），并仅报告哪些字符的频率是素数。

输入：输入的第一行给出一个整数T( 0<T<201)，表示测试用例个数。

后面的T行每行给出一个测试用例：一个字母-数字组成的字符串。

字符串的长度是小于2001的一个正整数。

输出：对输入的每个测试用例输出一行，给出一个输出序列号，然后给出在输入的字符串中频率是素数的字符。

这些字符按字母升序排列。

所谓“字母升序”意谓按ASCII 值升序排列。

如果没有字符的频率是素数，输出“empty”（没有引号）。

注：试题来源：Bangladesh National Computer Programming Contest在线测试：UV A 10789提示先离线计算出[2‥2200]的素数筛u[]。

然后每输入一个测试串，以ASCLL码为下标统计各字符的频率p[]，并按照ASCLL码递增的顺序（0≤i≤299）输出频率为素数的字符（即u [p[i]]=1且ASCLL码值为i的字符）。

若没有频率为素数的字符，则输出失败信息。

【2 Twin Primes】【问题描述】双素数（Twin Primes）是形式为(p, p+2)，术语“双素数”由Paul Stäckel (1892-1919)给出，前几个双素数是(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43)。

在本题中请你给出第S对双素数，其中S是输入中给出的整数。

复杂网络

Home Search Collections Journals About Contact us My IOPscienceMapping Koch curves into scale-free small-world networksThis article has been downloaded from IOPscience. Please scroll down to see the full text article.2010 J. Phys. A: Math. Theor. 43 395101(/1751-8121/43/39/395101)View the table of contents for this issue, or go to the journal homepage for moreDownload details:IP Address: 219.220.208.31The article was downloaded on 21/10/2010 at 07:26Please note that terms and conditions apply.IOP P UBLISHING J OURNAL OF P HYSICS A:M ATHEMATICAL AND T HEORETICAL J.Phys.A:Math.Theor.43(2010)395101(16pp)doi:10.1088/1751-8113/43/39/395101Mapping Koch curves into scale-free small-world networksZhongzhi Zhang1,2,Shuyang Gao1,2,Lichao Chen3,Shuigeng Zhou1,2,Hongjuan Zhang2,4and Jihong Guan51School of Computer Science,Fudan University,Shanghai200433,People’s Republic of China2Shanghai Key Lab of Intelligent Information Processing,Fudan University,Shanghai200433,People’s Republic of China3Electrical Engineering Department,University of California,Los Angeles,CA90024,USA4Department of Mathematics,College of Science,Shanghai University,Shanghai200444,People’s Republic of China5Department of Computer Science and Technology,Tongji University,4800Cao’an Road,Shanghai201804,People’s Republic of ChinaE-mail:zhangzz@,sgzhou@ and jhguan@Received8June2010,inﬁnal form7July2010Published1September2010Online at /JPhysA/43/395101AbstractThe class of Koch fractals is one of the most interesting families of fractals,andthe study of complex networks is a central issue in the scientiﬁc community.In this paper,inspired by the famous Koch fractals,we propose a mappingtechnique converting Koch fractals into a family of deterministic networkscalled Koch networks.This novel class of networks incorporates somekey properties characterizing a majority of real-life networked systems—apower-law distribution with exponent in the range between2and3,a highclustering coefﬁcient,a small diameter and average path length and degreecorrelations.Besides,we enumerate the exact numbers of spanning trees,spanning forests and connected spanning subgraphs in the networks.All thesefeatures are obtained exactly according to the proposed generation algorithm ofthe networks considered.The network representation approach could be usedto investigate the complexity of some real-world systems from the perspectiveof complex networks.PACS numbers:89.75.Hc,05.10.−a,89.75.Fb,61.43.Hv(Someﬁgures in this article are in colour only in the electronic version)1.IntroductionThe past decade has witnessed a great deal of activity devoted to complex networks by the scientiﬁc community,since many systems in the real world can be described and characterized 1751-8113/10/395101+16$30.00©2010IOP Publishing Ltd Printed in the UK&the USA1by complex networks[1–4].Prompted by the computerization of data acquisition and the increased computing power of computers,researchers have done a lot of empirical studies on diverse real networked systems,unveiling the presence of some generic properties of various natural and manmade networks:power-law degree distribution P(k)∼k−γwith the characteristic exponentγin the range between2and3[5],small-world effect including a large clustering coefﬁcient and small average distance[6],and degree correlations[7,8].The empirical studies have inspired researchers to construct network models with the aim to reproduce or explain the striking common features of real-life systems[1,2].In addition to the seminal Watts–Strogatz’s(WS)small-world network model[6]and Barab´a si–Albert’s (BA)scale-free network model[5],a considerable number of models and mechanisms have been developed to mimic real-world systems,including initial attractiveness[9],aging and cost[10],ﬁtness model[11],duplication[12],weight or trafﬁc driven evolution[13,14], geographical constraint[15],accelerating growth[16,17],coevolution[18]and visibility graph[19],to list a few.Although signiﬁcant progress has been made in theﬁeld of network modeling and has led to a signiﬁcant improvement in our understanding of complex systems,it is still a fundamental task and of current interest to construct models mimicking real networks and reproducing their generic properties from different angles[20].In this paper,enlightened by the famous class of Koch fractals,we propose a family of deterministic mathematical networks,called Koch networks,which integrates the observed properties of real networks in a single framework.We derive analytically exact scaling laws for degree distribution,clustering coefﬁcient,diameter,average distance or average path length (APL),degree correlations,even for spanning trees,spanning forests and connected spanning subgraphs.The obtained precise results show that Koch networks have rich topological features:they obey a power-law degree distribution with the exponent lies between2and3; they have a large clustering coefﬁcient and their diameter and APL grow logarithmically with the total number of nodes;and they may be either disassortative or uncorrelated.This work unfolds an alternative perspective in the study of complex networks.Instead of searching generation mechanisms for real networks,we explore deterministic mathematical networks that exhibit some typical properties of real-world systems.As the classical Koch fractals are important for the understanding of geometrical fractals in real systems[21],we believe that Koch networks could provide valuable insights into real-world systems.work constructionIn order to deﬁne the networks,weﬁrst introduce a classical fractal—Koch curve,which was proposed by von Koch[22].The Koch curve,denoted as S1(t)after t generations,can be constructed in a recursive way.To produce this well-known fractal,we begin with an equilateral triangle and let this initial conﬁguration be S1(0).In theﬁrst generation,we perform the following operations:ﬁrstly,we trisect each side of the initial equilateral triangle; secondly,on the middle segment of each side,we construct new equilateral triangles whose interiors lie external to the region enclosed by the base triangle;thirdly,we remove the three middle segments of the base triangle,upon which new triangles were established.Thus,we get S1(1).In the second generation,for each line segment in S1(1),we repeat the above procedure of three operations to obtain S1(2).This process is then repeated for successive generations. As t tends to inﬁnite,the Koch curve is obtained,and its Hausdorff dimension is d f=2ln2ln3 [23].Figure1depicts the structure of S1(2).The Koch curve can be easily generalized to other dimensions by introducing a parameter m(a positive integer)[23,24].The generalization after t generations is denoted by S m(t),which is constructed as follows[23]:start with an equilateral triangle as the initial conﬁguration 2Figure1.Theﬁrst two generations of the construction for the Koch curve.S m(0).In theﬁrst generation,we perform the following operations similar to those described in the last paragraph:partition each side of the initial triangle into2m+1segments,which are consecutively numbered1,2,...,2m,2m+1from one endpoint of the side to the other; construct a new small equilateral triangle on each even-numbered segment so that the interiors of the new triangles lie in the exterior of the base triangle;remove the segments upon which triangles were constructed.In this way we obtain S m(1).Analogously,we can get S m(t) from S m(t−1)by repeating recursively the procedure of the above three operations for each existing line segment in generation t−1.In the inﬁnite t limit,the Hausdorff dimension ofthe generalized Koch curves d f=ln(4m+1)ln(2m+1)[23].Figure2shows the structure of S2(2).The generalized Koch curves can be used as a basis of a new class of networks: sides(excluding those deleted)of the triangles of the Koch curves constructed at arbitrary generations are mapped to nodes,which are connected to one another if their corresponding sides in the Koch curves are in contact.For uniformity,the three sides of the initial equilateral triangle of S m(0)also correspond to three different nodes.We shall call the resultant networks Koch networks.Note that after establishing each side of a triangle constructed at a given generation of the Koch curves,although some segments of it will be removed at subsequent steps,we look at its remaining segments as a whole and map it to only one node.Figures3 and4show two networks corresponding to S1(2)and S2(2),respectively.Obviously,Koch networks have an inﬁnite number of nodes.But in what follows we shall generally consider the network characteristics after aﬁnite number of generations in the development of complete Koch networks.From our analytical results,we can quickly obtain the characteristics of the complete networks by taking the limit of large t.However,the numerical results are necessarily limited to networks withﬁnite order(number of all nodes).3.Generation algorithmAccording to the construction process of the generalized Koch curves and the proposed method of mapping from Koch curves to Koch networks,we can introduce an iterative algorithm with3Figure2.Theﬁrst two generations of the construction for the generalized Koch curve in the caseof m=2.Figure3.The network derived from S1(2).ease to create Koch networks,denoted by K m,t after t generation evolutions.The algorithm is as follows:initially(t=0),K m,0consists of three nodes forming a triangle.For t 1, K m,t is obtained from K m,t−1by adding m groups of nodes for each of the three nodes of every existing triangle in K m,t−1.Each node group has two nodes.These two new nodes and their‘mother’nodes are linked to one another shaping a new triangle.In other words,to obtain K m,t from K m,t−1,we replace each of the existing triangles of K m,t−1by the connected clusters on the rightmost side ofﬁgure5.Figures3and4illustrate the growing process of the networks for two particular cases of m=1and m=2,respectively.Note that in the peculiar case of m=1,the networks under consideration reduce to the one previously studied in[25].Let us compute the order and size(number of all edges)of the Koch networks K m,t.To this end,weﬁrst consider the total number of triangles L (t)that exist at step t.By construction 4Figure4.The network corresponding to S2(2).Figure5.Iterative construction method for the Koch networks.(seeﬁgure5),this quantity increases by a factor of3m+1,i.e.L (t)=(3m+1)L (t−1). Considering L (0)=1,we have L (t)=(3m+1)t.Denote L v(t)and L e(t)as the numbers of nodes and edges created at step t,respectively.Note that each triangle in K m,t−1 will give rise to6m new nodes and9m new edges at step t;then one can easily obtain L v(t)=6m L (t−1)=6m(3m+1)t−1and L e(t)=9m L (t−1)=9m(3m+1)t−1,both of which hold for arbitrary t>0.Then,the total numbers of nodes N t and edges E t present at step t areN t=tt i=0L v(t i)=2(3m+1)t+1(1)andE t=tt i=0L e(t i)=3(3m+1)t,(2)respectively.Thus,the average degree isk =2E tN t =6(3m+1)t2(3m+1)t+1,(3)5which is approximately3for large t,showing that Koch networks are sparse as most real-life networks[1–4].4.Topological propertiesNow we study some relevant characteristics of the Koch networks K m,t,focusing on degree distribution,clustering coefﬁcient,diameter,average distance,degree correlations,spanning trees,spanning forests and connected spanning subgraphs.We emphasize that this is the ﬁrst analytical study for counting spanning trees,spanning forests and connected spanning subgraphs in scale-free networks.4.1.Degree distributionLet k i(t)be the degree of node i at time t.When node i enters the network at step t i(t i 0), it has a degree of2,namely k i(t i)=2.To determine k i(t),weﬁrst consider the number of triangles involving node i at step t that is denoted by L (i,t).These triangles will give rise to new nodes linked to node i at step t+1.Then at step t i,L (i,t i)=1.By construction, for any triangle involving node i at a given step,it will lead to m new triangles passing by node i at a next step.Thus,L (i,t)=(m+1)L (i,t−1).Considering the initial condition L (i,t i)=1,we have L (i,t)=(m+1)t−t i.On the other hand,each triangle passing by node i contains two links connected to i;therefore,we have k i(t)=2L (i,t).Then we obtaink i(t)=2L (i,t)=2(m+1)t−t i.(4) In this way,at time t the degree of the arbitrary node i of Koch networks has been computed explicitly.From equation(4),it is easy to see that at each step the degree of a node increases m times,i.e.k i(t)=(m+1)k i(t−1).(5) Equation(4)shows that the degree spectrum of Koch networks is discrete.Thus,we can get the degree distribution P(k)of the Koch networks via the cumulative degree distribution [3]given byP cum(k)=1N tτ t iL v(τ)=2×(3m+1)t i+12×(3m+1)+1.(6)Substituting t i=t−ln(k2)ln(m+1)in this expression givesP cum(k)=2×(3m+1)t×k2−ln(3m+1)ln(m+1)+12×(3m+1)t+1.(7)In the inﬁnite t limit,we obtainP cum(k)=2ln(3m+1)ln(m+1)×k−ln(3m+1)ln(m+1).(8) So the degree distribution follows a power-law form P(k)∼k−γwith the exponentγ=1+ln(3m+1)ln(m+1)belonging to the interval[2,3].When m increases from1to inﬁnite,γdecreases from3to2.It should be stressed that the exponent of degree distribution of most real scale-free networks also lies in the same range between2and3.6Figure6.Semilogarithmic plot of the average clustering coefﬁcient C t versus the networkorder N t.4.2.Clustering coefﬁcientBy deﬁnition,the clustering coefﬁcient[6]of a node i with degree k i is the ratio between thenumber of triangles e i that actually exist among the k i neighbors of node i and the maximumpossible number of triangles involving i,k i(k i−1)/2,namely C i=2e i/[k i(k i−1)].For Koch networks,we can obtain the exact expression of the clustering coefﬁcient C(k)for asingle node with degree k.By construction,for any given node having a degree k,there are juste=k2triangles connected with this node;see also equation(4).Hence there is a one-to-onecorresponding relation between the clustering coefﬁcient of a node and its degree:for a node of degree k,C(k)=1k−1,(9)which shows a power-law scaling C(k)∼k−1in the large limit of k,in agreement with the behavior observed in a variety of real-life systems[26].After t step growth,the average clustering coefﬁcient C t of the whole network K m,t, deﬁned as the mean of Cis over all nodes in the network,is given byC t=1N ttr=01G r−1×L v(r),(10)where the sum runs over all the nodes of all generations and G r is the degree of those nodes created at step r,which is given by equation(4).In the limit of large N t,equation(10) converges to a nonzero value C,as reported inﬁgure6.For m=1,2and3,C is0.82008, 0.88271and0.91316,respectively.As m approaches inﬁnite,C converges to1.Thus,C increases with m:when m grows from1to inﬁnite,C increases form0.82008to1.Therefore, for the full range of m,the the average clustering coefﬁcient of Koch networks is very high.74.3.DiameterMost real networks are small-world,i.e.their average distance grows logarithmically with network order or slower.Here the average distance means the minimum number of edges connecting a pair of nodes,averaged over all node pairs.For a general network,it is not easy to derive a closed formula for its average distance.However,the whole family of Koch networks has a self-similar structure,allowing for analytically calculating the average distance,which approximately increases as a logarithmic function of the network order.We leave the detailed exact derivation about the average distance to the next subsection.Here we provide the exact result of the diameter of K m,t denoted by Diam(K m,t)for all parameters m,which is deﬁned as the maximum of the shortest distances between all pairs of nodes.Small diameter is consistent with the concept of small-world.The obtained diameter also scales logarithmically with the network order.The computation details are presented as follows.Clearly,at step t=0,Diam(K m,0)is equal to1.At each step t 1,we call newly created nodes at this step as active nodes.Since all active nodes are attached to those nodes existing in K m,t−1,so one can easily see that the maximum distance between any active node and those nodes in K m,t−1is not more than Diam(K m,t−1)+1and that the maximum distance between any pair of active nodes is at most Diam(K m,t−1)+2.Thus,at any step,the diameter of the network increases by2at most.Then we get2(t+1)as the diameter of Diam(K m,t). Equation(1)indicates that the logarithm of the order of Diam(K m,t)is proportional to t in the large limit t.Thus the diameter Diam(K m,t)grows logarithmically with the network order, showing that the Koch networks are small-world.4.4.Average path lengthUsing a method similar to but different from those in the literature[27,28],we now study analytically the average path length d t of the Koch networks K m,t.It follows thatd t=D tot(t)N t(N t−1)/2,(11)where D tot(t)is the total distance between all couples of nodes,i.e.D tot(t)=i∈K m,t,j∈K m,t,i=jd ij(t),(12)where d ij(t)is the shortest distance between nodes i and j in the networks K m,t.Note that Koch networks have a self-similar structure,which allows us to address D tot(t) analytically.This self-similar structure is obvious from an equivalent network construction method:to obtain K m,t,one can make3m+1copies of K m,t−1and join them at the hubs (namely nodes with largest degree).As shown inﬁgure7,the network K m,t+1may be obtained by the juxtaposition of3m+1copies of K m,t,which are labeled as K1m,t,K2m,t,...,K3m m,t,andK3m+1m,t ,respectively.We continue by exhibiting the procedure of the determination of the total distance and present the recurrence formula,which allows us to obtain D tot(t+1)of the t+1generation from D tot(t)of the t generation.From the obvious self-similar structure of Koch networks,it is easy to see that the total distance D tot(t+1)satisﬁes the recursion relationD tot(t+1)=(3m+1)D tot(t)+ t,(13) 8Figure7.Second construction method of Koch networks that highlights self-similarity.Thegraph after t+1construction steps,K m,t+1,consists of3m+1copies of K m,t denoted as Kθm,t(θ=1,2,3,...,3m,3m+1),which are connected to one another as above.where t is the sum over all shortest paths whose endpoints are not in the same Kθm,t branch. The solution of equation(13)isD tot(t)=(3m+1)t−1D tot(1)+t−1τ=1(3m+1)t−τ−1 τ.(14)All the paths contributing to t must go through at least one of the three edge nodes(i.e.the gray nodes X,Y and Z inﬁgure7)at which the different Kθm,t branches are connected.The analytical expression for t,called the length of crossing paths,is found below.Let α,βt be the sum of the lengths of all shortest paths with endpoints in Kαm,t and Kβm,t. Based on whether or not two branches are adjacent,we sort the crossing path length α,βt into two classes:if Kαm,t and Kβm,t meet at an edge node, α,βt rules out the paths where either endpoint is that shared edge node.For example,each path contributed to 1,2t should not end at node X.If Kαm,t and Kβm,t do not meet, α,βt excludes the paths where either endpoint is any edge node.For instance,each path contributed to 2,m+2tshould not end at node X or Y.We can easily compute that the numbers of the two types of crossing paths are3m2+3m2and3m2,respectively.On the other hand,any two crossing paths belonging to the same class have identical length.Thus,the total sum t is given byt=3m2+3m21,2t+3m2 2,m+2t.(15)In order to determine 1,2t and 2,m+2t,we deﬁnes t=i∈K m,t,i=Xd iX(t).(16)Considering the self-similar network structure,we can easily know that at time t+1,the quantity s t+1evolves recursively ass t+1=(m+1)s t+2m[s t+(N t−1)]=(3m+1)s t+4m(3m+1)t.(17)9Using s0=2,we haves t=(4mt+6m+2)(3m+1)t−1.(18) Having obtained s t,the next step is to compute the quantities 1,2t and 2,m+2tgiven by 1,2t=i∈K1m,t,j∈K2m,ti,j=Xd ij(t+1)=i∈K1m,t,j∈K2m,ti,j=X[d iX(t+1)+d jX(t+1)]=(N t−1)i∈K1m,ti=X d iX(t+1)+(N t−1)j∈K2m,tj=Xd jX(t+1)=2(N t−1)i∈K1m,ti=Xd iX(t+1)=2(N t−1)s t,(19) and2,m+2t =i∈K2m,t,i=Xj∈K m+2m,t,j=Yd ij(t+1)=i∈K2m,t,i=Xj∈K m+2m,t,j=Y[d iX(t+1)+d XY(t+1)+d jY(t+1)]=2(N t−1)s t+(N t−1)2,(20)where d XY(t+1)=1has been used.Substituting equations(19)and(20)into equation(15), we obtaint=(9m2+3m)(N t−1)s t+3m2(N t−1)2=12m(2mt+4m+1)(3m+1)2t.(21) Inserting equations(21)for τinto equation(14),and using D tot(1)=48m2+21m+3,we can exactly obtain the expression for D tot(t)asD tot(t)=(3m+1)t−13[3m+5+(24mt+24m+4)(3m+1)t].(22)By inserting equation(22)into equation(11),one can obtain the analytical expression for d t:d t=3m+5+(24mt+24m+4)(3m+1)t3(3m+1)[2(3m+1)t+1],(23)which approximates4mt3m+1in the inﬁnite t,implying that the APL shows a logarithmic scalingwith network order.This again shows that the Koch networks exhibit a small-world behavior. We have checked our analytic result for d t given in equation(23)against numerical calculations for different m and various t.In all the cases we obtain complete agreement between our theoretical formula and the results of numerical investigation,seeﬁgure8.10Figure8.Average path length d t versus network order N t on a semi-log scale.The solid lines areguides to the eyes.4.5.Degree correlationsDegree correlation is a particularly interesting subject in theﬁeld of network science[7,8, 29–32]because it can give rise to some interesting network structure effects.An interesting quantity related to degree correlations is the average degree of the nearest neighbors for nodes with degree k,denoted as k nn(k),which is a function of the node degree k[30,31].When k nn(k)increases with k,it means that nodes have a tendency to connect to the nodes with a similar or larger degree.In this case the network is deﬁned as assortative[7,8].In contrast, if k nn(k)is decreasing with k,which implies that the nodes of large degree are likely to have near neighbors with small degree,then the network is said to be disassortative.If correlations are absent,k nn(k)=const.We can exactly calculate k nn(k)for Koch networks using equations(4)and(5)to work out how many links are made at a particular step to nodes with a particular degree.By construction,we have the following expression:k nn(k)=1L v(t i)k(t i,t)ti=t i−1ti=0m L v(t i)k(t i,t i−1)k(t i,t)+ti=tti=t i+1m L v(t i)k(t i,t i−1)k(t i,t)+1(24)for k=2(m+1)t−t i.Here theﬁrst sum on the right-hand side accounts for the links made tonodes with a larger degree(i.e.ti <t i)when the node was generated at t i.The second sumdescribes the links made to the current smallest degree nodes at each step ti >t i.The last term1accounts for the link connected to the simultaneously emerging node.In order to compute equation(24),we distinguish two cases according to the parameter m:m=1and m 2.When m=1,we havek nn(k)=t+2.(25)11Thus,in the case of m=1,the networks show the absence of correlations in the full range of t.From equations(25)and(1)we can easily see that for large t,k nn(k)is approximately a logarithmic function of the network order N t,namely k nn(k)∼ln N t,exhibiting a similar behavior as that of the BA model[31]and the two-dimensional random Apollonian network [32].When m 2,equation(24)is simpliﬁed tok nn(k)=3m+1)m−1(m+1)23m+1t i−m+3m−1+2mm+1(t−t i).(26)Thus after the initial step k nn(k)grows linearly with time.Writing equation(26)in terms of k,it is straightforward to obtaink nn(k)=3m+1m−1(m+1)23m+1tk2−ln[(m+1)23m+1]ln(m+1)−m+3m−1+2mm+1lnk2ln(m+1).(27)Therefore,k nn(k)is approximately a power-law function of k with negative exponent,which shows that the networks are disassortative.Note that k nn(k)of the Internet exhibits a similar power-law dependence on the degree k nn(k)∼k−ω,withω=0.5[30].4.6.Spanning trees,spanning forests and connected spanning subgraphsSpanning trees,spanning forests and connected spanning subgraphs are important quantities of networks,and the enumeration of these interesting quantities in networks is a fundamental issue[33–37].However,explicitly determining the numbers of these quantities in networks is a theoretical challenge[38].Fortunately,the peculiar construction of Koch networks makes it possible to derive exactly the three variables.4.6.1.Spanning trees.By deﬁnition,a spanning tree of any connected network is a minimal set of edges that connect every node.The problem of spanning trees is closely related to various aspects of networks,such as reliability[39,40],optimal synchronization[41]and random walks[42].Thus,it is of great interest to determine the exact number of spanning trees[43].In what follows we will examine the number of spanning trees in Koch networks.Note that in the Koch networks K m,t there are L (t)=(3m+1)t triangles,but there are no cycles of length more than3.For each of L (t)=(3m+1)t triangles,to assure that its three nodes are in one tree,only two edges of it must be present.Obviously,there are three possibilities for this.Thus,the total number of spanning trees in K m,t,denoted by N ST(t),isN ST(t)=3L (t)=3(3m+1)t.(28) We proceed to represent N ST(t)as a function of the network order N t,with the aim to provide the relation governing the two quantities.From equation(1),we have(3m+1)t=N t−12. This expression allows one to write N ST(t)in terms of N t asN ST(t)=3(N t−1)/2.(29)Thus,the number of spanning trees in K m,t increases exponentially with the network order N t,which means that there exists a constant E ST,called as the entropy of spanning trees, describing this exponential growth[34]:E ST=limN t→∞ln N ST(t)N t=12ln3.(30)In addition to the above analytical computation,according to the previously known result [44],one can also obtain numerically but exactly the number of spanning trees,N ST(t),by 12computing the nonzero eigenvalues of the Laplacian matrix associated with the networks K m,t asN ST(t)=1N ti=N t−1i=1λi(t),(31)whereλi(t)(i=1,2,...,N t−1)are the N t−1nonzero eigenvalues of the Laplacian matrix, denoted by L t,for the networks K m,t,which is deﬁned as follows:its non-diagonal element l ij(t)(i=j)is−1(or0)if nodes i and j are(or not)directly linked to each other,while the diagonal entry l ii(t)is exactly the degree of node i.Using equation(31),we have calculated directly the number of spanning trees in the networks K m,t,and the results from equation(31)are fully consistent with those obtained from equation(28),showing that our analytical formula is right.It should be stressed that although expression(31)seems compact,it is involved in the computation of the eigenvalues of a matrix of order N t×N t,which makes heavy demands on time and computational resources. Thus,it is not acceptable for large networks.In particular,by virtue of the eigenvalue method it is difﬁcult and even impossible to obtain the entropy E ST.Our analytical computation can get around the two difﬁculties,but is only applicable to peculiar networks.4.6.2.Spanning forests.To deﬁne spanning forests,weﬁrst recall the deﬁnition for a spanning subgraph.A spanning subgraph of a network is a subgraph having the same node as of the network but having partial or all edges of the original graph.A spanning forest of a network is a spanning graph of it that is a disjoint union of trees(here an isolated node is consider as a tree),i.e.a spanning graph without any cycle.The enumeration of spanning forests is very interesting since it corresponds to the partition function of the q-state Potts model[45]in the limit of q→0.For a general network,it is very hard to count the number of its spanning subgraphs.But below we will show that for the Koch networks K m,t,the number of spanning subgraphs,N SF(t),can be obtained explicitly.Analogous to the enumeration of spanning trees,for each triangle in K m,t,to guarantee the absence of cycle among its three nodes,at least one edge must be removed.And there are total seven possibilities for deleting the edges of a triangle.Then the number of spanning forests in K m,t isN SF(t)=7L (t)=7(3m+1)t,(32) which can be rewritten as a function of the network order N t asN SF(t)=7(N t−1)/2.(33) Therefore,N SF(t)also grows exponentially in N t,which allows for deﬁning the entropy of the spanning forests of Koch networks as the limiting value[46]:E SF=limN t→∞ln N SF(t)N t=12ln7.(34)Thus,we have obtained the rigorous results for the number of spanning forests in Koch networks and its entropy.4.6.3.Connected spanning subgraphs.As the name suggests,a connected spanning subgraph of a connected network is a spanning subgraph of the network,which remains connected.By applying a method similar to that given above,we can compute the number of connected spanning subgraphs in the Koch networks K m,t,which is denoted by N CSS(t).For any triangle in K m,t,to ensure the connectedness of its three nodes,at most one edge can be13。

ImageProcessing_Unit1_slides爱丁堡大学图像处理研究生课程课件

• • • • No noise Not blurred High resolution Good contrast
Image Processing - Unit 1
javier.escudero@
10
Institute for Digital Communications (IDCOM) – Image Processing
256 × 256 pixels 128 × 128 pixels
Fig. 1.6 in Petrou’s
Image Processing - Unit 1 javier.escudero@ 12
Institute for Digital Communications (IDCOM) – Image Processing
Institute for Digital Communications (IDCOM) – Image Processing
Unit 1 – Introduction to digital image processing
Dr Javier Escudero javier.escudero@ School of Engineering / University of Edinburgh
False contouring – Content effects
Fig. 1.6 in Petrou’s
Image Processing - Unit 1 javier.escudero@ 17
Institute for Digital Communications (IDCOM) – Image Processing
False contouring (I)
• Keeping the size of the image constant and reducing the number of grey levels (G=2m) produces false contouring • However, this effect depends on the contents of the image

POJ题目分类

1000 A+B Problem 送分题1001 Exponentiation 高精度1003 Hangover 送分题1004 Financial Management 送分题1005 I Think I Need a Houseboat 几何1006 Biorhythms 送分题1007 DNA Sorting 送分题1008 Maya Calendar 日期处理1010 STAMPS 搜索＋DP1011 Sticks 搜索1012 Joseph 模拟/数学方法1014 Dividing 数论/DP?/组合数学->母函数?1015 Jury Compromise DP1016 Numbers That Count 送分题1017 Packets 贪心1018 Communication System 贪心1019 Number Sequence 送分题1020 Anniversary Cake 搜索1023 The Fun Number System 数论1025 Department 模拟1026 Cipher 组合数学1027 The Same Game 模拟1028 Web Navigation 送分题1031 Fence 计算几何1034 The dog task 计算几何1037 A decorative fence DP/组合数学1039 Pipe 几何1042 Gone Fishing 贪心/DP1045 Bode Plot 送分题(用物理知识)1046 Color Me Less 送分题1047 Round and Round We Go 高精度1048 Follow My Logic 模拟1049 Microprocessor Simulation 模拟1050 To the Max DP1053 Set Me 送分题1054 The Troublesome Frog 搜索1060 Modular multiplication of polynomials 高精度1061 青蛙的约会数论1062 昂贵的聘礼DP1064 Cable master DP/二分查找1065 Wooden Sticks DP1067 取石子游戏博弈论1068 Parencodings 送分题1069 The Bermuda Triangle 搜索1070 Deformed Wheel 几何1071 Illusive Chase 送分题1072 Puzzle Out 搜索1073 The Willy Memorial Program 模拟1074 Parallel Expectations DP1075 University Entrance Examination 模拟1080 Human Gene Functions DP->LCS变形1082 Calendar Game 博弈论1084 Square Destroyer 搜索？1085 Triangle War 博弈论1086 Unscrambling Images 模拟?1087 A Plug for UNIX 图论->最大流1088 滑雪DFS/DP1090 Chain ->格雷码和二进制码的转换1091 跳蚤数论1092 Farmland 几何1093 Formatting Text DP1094 Sorting It All Out 图论->拓扑排序1095 Trees Made to Order 组合数学1096 Space Station Shielding 送分题1097 Roads Scholar 图论1098 Robots 模拟1099 Square Ice 送分题1100 Dreisam Equations 搜索1101 The Game 搜索->BFS1102 LC-Display 送分题1103 Maze 模拟1104 Robbery 递推1106 Transmitters 几何1107 W's Cipher 送分题1110 Double Vision 搜索1111 Image Perimeters 搜索1112 Team Them Up! DP1113 Wall 计算几何->convex hull1119 Start Up the Startup 送分题1120 A New Growth Industry 模拟1122 FDNY to the Rescue! 图论->Dijkstra 1125 Stockbroker Grapevine 图论->Dijkstra 1128 Frame Stacking 搜索1129 Channel Allocation 搜索（图的最大独立集）1131 Octal Fractions 高精度1135 Domino Effect 图论->Dijkstra1137 The New Villa 搜索->BFS1141 Brackets Sequence DP1142 Smith Numbers 搜索1143 Number Game 博弈论1147 Binary codes 构造1148 Utopia Divided 构造1149 PIGS 图论->网络流1151 Atlantis 计算几何->同等安置矩形的并的面积->离散化1152 An Easy Problem! 数论1157 LITTLE SHOP OF FLOWERS DP1158 TRAFFIC LIGHTS 图论->Dijkstra变形1159 Palindrome DP->LCS1160 Post Office DP1161 Walls 图论1162 Building with Blocks 搜索1163 The Triangle DP1170 Shopping Offers DP1177 Picture 计算几何->同等安置矩形的并的周长->线段树1179 Polygon DP1180 Batch Scheduling DP1182 食物链数据结构->并查集1183 反正切函数的应用搜索1184 聪明的打字员搜索1185 炮兵阵地DP->数据压缩1187 陨石的秘密DP（BalkanOI99 Par的拓展）1189 钉子和小球递推?1190 生日蛋糕搜索/DP1191 棋盘分割DP1192 最优连通子集图论->无负权回路的有向图的最长路->BellmanFord 1193 内存分配模拟1194 HIDDEN CODES 搜索+DP1197 Depot 数据结构->Young T ableau1201 Intervals 贪心/图论->最长路->差分约束系统1202 Family 高精度1209 Calendar 日期处理1217 FOUR QUARTERS 递推1218 THE DRUNK JAILER 送分题1233 Street Crossing 搜索->BFS1245 Programmer, Rank Thyself 送分题1247 Magnificent Meatballs 送分题1248 Safecracker 搜索1250 T anning Salon 送分题1251 Jungle Roads 图论->最小生成树1271 Nice Milk 计算几何1273 Drainage Ditches 图论->最大流1274 The Perfect Stall 图论->二分图的最大匹配1275 Cashier Employment 图论->差分约束系统->无负权回路的有向图的最长路->Bellman-Ford1280 Game 递推1281 MANAGER 模拟1286 Necklace of Beads 组合数学->Polya定理1288 Sly Number 数论->解模线性方程组1293 Duty Free Shop DP1298 The Hardest Problem Ever 送分题1316 Self Numbers 递推同Humble Number一样1322 Chocolate 递推/组合数学1323 Game Prediction 贪心1324 Holedox Moving BFS+压缩储存1325 Machine Schedule 图论->二分图的最大匹配1326 Mileage Bank 送分题1327 Moving Object Recognition 模拟?1328 Radar Installation 贪心（差分约束系统的特例）1338 Ugly Numbers 递推(有O(n)算法)1364 King 图论->无负权回路的有向图的最长路->BellmanFord1370 Gossiping (数论->模线性方程有无解的判断)+(图论->DFS)2184 Cow Exhibition DP2190 ISBN 送分题2191 Mersenne Composite Numbers 数论2192 Zipper DP->LCS变形2193 Lenny's Lucky Lotto Lists DP2194 Stacking Cylinders 几何2195 Going Home 图论->二分图的最大权匹配2196 Specialized Four-Digit Numbers 送分题2197 Jill's Tour Paths 图论->2199 Rate of Return 高精度2200 A Card Trick 模拟2210 Metric Time 日期处理2239 Selecting Courses 图论->二分图的最大匹配2243 Knight Moves 搜索->BFS2247 Humble Numbers 递推(最优O(n)算法)2253 Frogger 图论->Dijkstra变形(和1295是一样的)2254 Globetrotter 几何2261 France '98 递推2275 Flipping Pancake 构造2284 That Nice Euler Circuit 计算几何2289 Jamie's Contact Groups 图论->网络流?2291 Rotten Ropes 送分题2292 Optimal Keypad DP2299 Ultra-QuickSort 排序->归并排序2304 Combination Lock 送分题2309 BST 送分题2311 Cutting Game 博弈论2312 Battle City 搜索->BFS2314 POJ language 模拟2315 Football Game 几何2346 Lucky tickets 组合数学2351 Time Zones 时间处理2379 ACM Rank T able 模拟+排序2381 Random Gap 数论2385 Apple Catching DP(像NOI98“免费馅饼”)2388 Who's in the Middle 送分题(排序)2390 Bank Interest 送分题2395 Out of Hay 图论->Dijkstra变形2400 Supervisor, Supervisee 图论->二分图的最大权匹配?2403 Hay Points 送分题2409 Let it Bead 组合数学->Polya定理2416 Return of the Jedi 图论->2417 Discrete Logging 数论2418 Hardwood Species 二分查找2419 Forests 枚举2421 Constructing Roads 图论->最小生成树2423 The Parallel Challenge Ballgame 几何2424 Flo's Restaurant 数据结构->堆2425 A Chess Game 博弈论2426 Remainder BFS2430 Lazy Cows DP->数据压缩1375 Intervals 几何1379 Run Away 计算几何->1380 Equipment Box 几何1383 Labyrinth 图论->树的最长路1394 Railroad 图论->Dijkstra1395 Cog-Wheels 数学->解正系数的线性方程组1408 Fishnet 几何1411 Calling Extraterrestrial Intelligence Again 送分题1430 Binary Stirling Numbers 日期处理1431 Calendar of Maya 模拟1432 Decoding Morse Sequences DP1434 Fill the Cisterns! 计算几何->离散化/1445 Random number 数据结构->碓1447 Ambiguous Dates 日期处理1450 Gridland 图论(本来TSP问题是NP难的，但这个图比较特殊，由现成的构造方法)1458 Common Subsequence DP->LCS1459 Power Network 图论->最大流1462 Random Walk 模拟+解线性方程组1463 Strategic game 贪心1466 Girls and Boys 图论->n/a1469 COURSES 贪心1475 Pushing Boxes DP1476 Always On the Run 搜索->BFS1480 Optimal Programs 搜索->BFS1481 The Die Is Cast 送分题1482 It's not a Bug, It's a Feature! 搜索->BFS1483 Going in Circles on Alpha Centauri 模拟1484 Blowing Fuses 送分题1485 Fast Food DP(似乎就是ioi2000的postoffice)1486 Sorting Slides 图论->拓扑排序1505 Copying Books DP+二分查找1510 Hares and Foxes 数论1512 Keeps Going and Going and ... 模拟1513 Scheduling Lectures DP1514 Metal Cutting 几何1515 Street Directions 图论->把一个无向连通图改造成为有向强连通图1517 u Calculate e 送分题1518 Problem Bee 几何1519 Digital Roots 送分题(位数可能很大)1520 Scramble Sort 排序1547 Clay Bully 送分题1555 Polynomial Showdown 送分题(非常阴险)1563 The Snail 送分题1601 Pizza Anyone? 搜索1604 Just the Facts 送分题1605 Horse Shoe Scoring 几何1606 Jugs 数论/搜索1631 Bridging signals DP+二分查找1632 Vase collection 图论->最大完全图1633 Gladiators DP1634 Who's the boss? 排序1635 Subway tree systems 图论->不同表示法的二叉树判同1637 Sightseeing tour 图论->欧拉回路1638 A number game 博弈论1639 Picnic Planning 图论->1641 Rational Approximation 数论1646 Double Trouble 高精度1654 Area 几何1657 Distance on Chessboard 送分题1658 Eva's Problem 送分题1660 Princess FroG 构造1661 Help Jimmy DP1663 Number Steps 送分题1664 放苹果组合数学->递推1677 Girls' Day 送分题1688 Dolphin Pool 计算几何1690 (Your)((Term)((Project))) 送分题1691 Painting A Board 搜索/DP1692 Crossed Matchings DP1693 Counting Rectangles 几何1694 An Old Stone Game 博弈论?1695 Magazine Delivery 图论->1712 Flying Stars DP1713 Divide et unita 搜索1714 The Cave 搜索/DP1717 Dominoes DP1718 River Crossing DP1719 Shooting Contest 贪心1729 Jack and Jill 图论->1730 Perfect Pth Powers 数论1732 Phone numbers DP1734 Sightseeing trip 图论->Euler回路1738 An old Stone Game 博弈论?1741 Tree 博弈论?1745 Divisibility DP1751 Highways 图论->1752 Advertisement 贪心/图论->差分约束系统1753 Flip Game 搜索->BFS1755 Triathlon 计算几何?1770 Special Experiment 树形DP1771 Elevator Stopping Plan DP1772 New Go Game 构造?1773 Outernet 模拟1774 Fold Paper Strips 几何1775 Sum of Factorials 送分题1776 T ask Sequences DP1777 Vivian's Problem 数论1870 Bee Breeding 送分题1871 Bullet Hole 几何1872 A Dicey Problem BFS1873 The Fortified Forest 几何+回溯1874 Trade on Verweggistan DP1875 Robot 几何1876 The Letter Carrier's Rounds 模拟1877 Flooded! 数据结构->堆1879 Tempus et mobilius Time and motion 模拟+组合数学->Polya定理1882 Stamps 搜索+DP1883 Theseus and the Minotaur 模拟1887 Testing the CATCHER DP1889 Package Pricing DP1893 Monitoring Wheelchair Patients 模拟+几何1915 Knight Moves 搜索->BFS1916 Rat Attack 数据结构->?1936 All in All DP?1946 Cow Cycling DP1947 Rebuilding Roads 二分1985 Cow Marathon 图论->有向无环图的最长路1995 Raising Modulo Numbers 数论->大数的幂求余2049 Finding Nemo 图论->最短路2050 Searching the Web 模拟(需要高效实现)2051 Argus 送分题(最好用堆，不用也可以过)2054 Color a Tree 贪心2061 Pseudo-random Numbers 数论2080 Calendar 日期处理2082 Terrible Sets 分治/2083 Fractal 递归2084 Game of Connections 递推(不必高精度)2105 IP Address 送分题2115 C Looooops 数论->解模线性方程2136 Vertical Histogram 送分题2165 Gunman 计算几何2179 Inlay Cutters 枚举2181 Jumping Cows 递推2182 Lost Cows ->线段树/＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝＝1370 Gossiping (数论->模线性方程有无解的判断)+(图论->DFS)1090 Chain ->格雷码和二进制码的转换2182 Lost Cows ->线段树/2426 Remainder BFS1872 A Dicey Problem BFS1324 Holedox Moving BFS+压缩储存1088 滑雪DFS/DP1015 Jury Compromise DP1050 To the Max DP1062 昂贵的聘礼DP1065 Wooden Sticks DP1074 Parallel Expectations DP1093 Formatting Text DP1112 Team Them Up! DP1141 Brackets Sequence DP1157 LITTLE SHOP OF FLOWERS DP1160 Post Office DP1163 The Triangle DP1170 Shopping Offers DP1179 Polygon DP1180 Batch Scheduling DP1191 棋盘分割DP1293 Duty Free Shop DP2184 Cow Exhibition DP2193 Lenny's Lucky Lotto Lists DP2292 Optimal Keypad DP1432 Decoding Morse Sequences DP1475 Pushing Boxes DP1513 Scheduling Lectures DP1633 Gladiators DP1661 Help Jimmy DP1692 Crossed Matchings DP1712 Flying Stars DP1717 Dominoes DP1718 River Crossing DP1732 Phone numbers DP1745 Divisibility DP1771 Elevator Stopping Plan DP1776 T ask Sequences DP1874 Trade on Verweggistan DP1887 Testing the CATCHER DP1889 Package Pricing DP1946 Cow Cycling DP1187 陨石的秘密DP（BalkanOI99 Par的拓展）1485 Fast Food DP(似乎就是ioi2000的postoffice) 2385 Apple Catching DP(像NOI98“免费馅饼”) 1064 Cable master DP/二分查找1037 A decorative fence DP/组合数学1936 All in All DP?1505 Copying Books DP+二分查找1631 Bridging signals DP+二分查找1159 Palindrome DP->LCS1458 Common Subsequence DP->LCS1080 Human Gene Functions DP->LCS变形2192 Zipper DP->LCS变形1185 炮兵阵地DP->数据压缩2430 Lazy Cows DP->数据压缩1067 取石子游戏博弈论1082 Calendar Game 博弈论1085 Triangle War 博弈论1143 Number Game 博弈论2311 Cutting Game 博弈论2425 A Chess Game 博弈论1638 A number game 博弈论1694 An Old Stone Game 博弈论?1738 An old Stone Game 博弈论?1741 Tree 博弈论?2083 Fractal 递归1104 Robbery 递推1217 FOUR QUARTERS 递推1280 Game 递推2261 France '98 递推2181 Jumping Cows 递推1316 Self Numbers 递推同Humble Number一样2084 Game of Connections 递推(不必高精度) 1338 Ugly Numbers 递推(有O(n)算法)2247 Humble Numbers 递推(最优O(n)算法)1322 Chocolate 递推/组合数学1189 钉子和小球递推?1947 Rebuilding Roads 二分2418 Hardwood Species 二分查找2082 Terrible Sets 分治/1001 Exponentiation 高精度1047 Round and Round We Go 高精度1060 Modular multiplication of polynomials 高精度1131 Octal Fractions 高精度1202 Family 高精度2199 Rate of Return 高精度1646 Double Trouble 高精度1147 Binary codes 构造1148 Utopia Divided 构造2275 Flipping Pancake 构造1660 Princess FroG 构造1772 New Go Game 构造?1005 I Think I Need a Houseboat 几何1039 Pipe 几何1070 Deformed Wheel 几何1092 Farmland 几何1106 Transmitters 几何2194 Stacking Cylinders 几何2254 Globetrotter 几何2315 Football Game 几何2423 The Parallel Challenge Ballgame 几何1375 Intervals 几何1380 Equipment Box 几何1408 Fishnet 几何1514 Metal Cutting 几何1518 Problem Bee 几何1605 Horse Shoe Scoring 几何1654 Area 几何1693 Counting Rectangles 几何1774 Fold Paper Strips 几何1871 Bullet Hole 几何1875 Robot 几何1873 The Fortified Forest 几何+回溯1031 Fence 计算几何1034 The dog task 计算几何1271 Nice Milk 计算几何2284 That Nice Euler Circuit 计算几何1688 Dolphin Pool 计算几何2165 Gunman 计算几何1755 Triathlon 计算几何?1379 Run Away 计算几何->1113 Wall 计算几何->convex hull1434 Fill the Cisterns! 计算几何->离散化/1151 Atlantis 计算几何->同等安置矩形的并的面积->离散化1177 Picture 计算几何->同等安置矩形的并的周长->线段树2419 Forests 枚举2179 Inlay Cutters 枚举1025 Department 模拟1027 The Same Game 模拟1048 Follow My Logic 模拟1049 Microprocessor Simulation 模拟1073 The Willy Memorial Program 模拟1075 University Entrance Examination 模拟1098 Robots 模拟1103 Maze 模拟1120 A New Growth Industry 模拟1193 内存分配模拟1281 MANAGER 模拟2200 A Card Trick 模拟2314 POJ language 模拟1431 Calendar of Maya 模拟1483 Going in Circles on Alpha Centauri 模拟1512 Keeps Going and Going and ... 模拟1773 Outernet 模拟1876 The Letter Carrier's Rounds 模拟1883 Theseus and the Minotaur 模拟2050 Searching the Web 模拟(需要高效实现)1012 Joseph 模拟/数学方法1086 Unscrambling Images 模拟?1327 Moving Object Recognition 模拟?1893 Monitoring Wheelchair Patients 模拟+几何1462 Random Walk 模拟+解线性方程组2379 ACM Rank T able 模拟+排序1879 Tempus et mobilius Time and motion 模拟+组合数学->Polya定理1520 Scramble Sort 排序1634 Who's the boss? 排序2299 Ultra-QuickSort 排序->归并排序1008 Maya Calendar 日期处理1209 Calendar 日期处理2210 Metric Time 日期处理1430 Binary Stirling Numbers 日期处理1447 Ambiguous Dates 日期处理2080 Calendar 日期处理2351 Time Zones 时间处理1770 Special Experiment 树形DP1916 Rat Attack 数据结构->?1197 Depot 数据结构->Young T ableau1182 食物链数据结构->并查集2424 Flo's Restaurant 数据结构->堆1877 Flooded! 数据结构->堆1445 Random number 数据结构->碓1023 The Fun Number System 数论1061 青蛙的约会数论1091 跳蚤数论1152 An Easy Problem! 数论2191 Mersenne Composite Numbers 数论2381 Random Gap 数论2417 Discrete Logging 数论1510 Hares and Foxes 数论1641 Rational Approximation 数论1730 Perfect Pth Powers 数论1777 Vivian's Problem 数论2061 Pseudo-random Numbers 数论1014 Dividing 数论/DP?/组合数学->母函数?1606 Jugs 数论/搜索1995 Raising Modulo Numbers 数论->大数的幂求余2115 C Looooops 数论->解模线性方程1288 Sly Number 数论->解模线性方程组1395 Cog-Wheels 数学->解正系数的线性方程组1000 A+B Problem 送分题1003 Hangover 送分题1004 Financial Management 送分题1006 Biorhythms 送分题1007 DNA Sorting 送分题1016 Numbers That Count 送分题1019 Number Sequence 送分题1028 Web Navigation 送分题1046 Color Me Less 送分题1053 Set Me 送分题1068 Parencodings 送分题1071 Illusive Chase 送分题1096 Space Station Shielding 送分题1099 Square Ice 送分题1102 LC-Display 送分题1107 W's Cipher 送分题1119 Start Up the Startup 送分题1218 THE DRUNK JAILER 送分题1245 Programmer, Rank Thyself 送分题1247 Magnificent Meatballs 送分题1250 T anning Salon 送分题1298 The Hardest Problem Ever 送分题1326 Mileage Bank 送分题2190 ISBN 送分题2196 Specialized Four-Digit Numbers 送分题2291 Rotten Ropes 送分题2304 Combination Lock 送分题2309 BST 送分题2390 Bank Interest 送分题2403 Hay Points 送分题1411 Calling Extraterrestrial Intelligence Again 送分题1481 The Die Is Cast 送分题1484 Blowing Fuses 送分题1517 u Calculate e 送分题1547 Clay Bully 送分题1563 The Snail 送分题1604 Just the Facts 送分题1657 Distance on Chessboard 送分题1658 Eva's Problem 送分题1663 Number Steps 送分题1677 Girls' Day 送分题1690 (Your)((Term)((Project))) 送分题1775 Sum of Factorials 送分题1870 Bee Breeding 送分题2105 IP Address 送分题2136 Vertical Histogram 送分题1555 Polynomial Showdown 送分题(非常阴险) 2388 Who's in the Middle 送分题(排序)1519 Digital Roots 送分题(位数可能很大)1045 Bode Plot 送分题(用物理知识)2051 Argus 送分题(最好用堆，不用也可以过) 1011 Sticks 搜索1020 Anniversary Cake 搜索1054 The Troublesome Frog 搜索1069 The Bermuda Triangle 搜索1072 Puzzle Out 搜索1100 Dreisam Equations 搜索1110 Double Vision 搜索1111 Image Perimeters 搜索1128 Frame Stacking 搜索1142 Smith Numbers 搜索1162 Building with Blocks 搜索1183 反正切函数的应用搜索1184 聪明的打字员搜索1248 Safecracker 搜索1601 Pizza Anyone? 搜索1713 Divide et unita 搜索1129 Channel Allocation 搜索（图的最大独立集）1190 生日蛋糕搜索/DP1691 Painting A Board 搜索/DP1714 The Cave 搜索/DP1084 Square Destroyer 搜索？1010 STAMPS 搜索＋DP1194 HIDDEN CODES 搜索+DP1882 Stamps 搜索+DP1101 The Game 搜索->BFS1137 The New Villa 搜索->BFS1233 Street Crossing 搜索->BFS2243 Knight Moves 搜索->BFS2312 Battle City 搜索->BFS1476 Always On the Run 搜索->BFS1480 Optimal Programs 搜索->BFS1482 It's not a Bug, It's a Feature! 搜索->BFS 1753 Flip Game 搜索->BFS1915 Knight Moves 搜索->BFS1017 Packets 贪心1018 Communication System 贪心1323 Game Prediction 贪心1463 Strategic game 贪心1469 COURSES 贪心1719 Shooting Contest 贪心2054 Color a Tree 贪心1328 Radar Installation 贪心（差分约束系统的特例）1042 Gone Fishing 贪心/DP1752 Advertisement 贪心/图论->差分约束系统1201 Intervals 贪心/图论->最长路->差分约束系统1097 Roads Scholar 图论1161 Walls 图论1450 Gridland 图论(本来TSP问题是NP难的，但这个图比较特殊，由现成的构造方法)2197 Jill's Tour Paths 图论->2416 Return of the Jedi 图论->1639 Picnic Planning 图论->1695 Magazine Delivery 图论->1729 Jack and Jill 图论->1751 Highways 图论->1122 FDNY to the Rescue! 图论->Dijkstra1125 Stockbroker Grapevine 图论->Dijkstra1135 Domino Effect 图论->Dijkstra1394 Railroad 图论->Dijkstra1158 TRAFFIC LIGHTS 图论->Dijkstra变形2395 Out of Hay 图论->Dijkstra变形2253 Frogger 图论->Dijkstra变形(和1295是一样的)1734 Sightseeing trip 图论->Euler回路1466 Girls and Boys 图论->n/a1515 Street Directions 图论->把一个无向连通图改造成为有向强连通图1635 Subway tree systems 图论->不同表示法的二叉树判同1275 Cashier Employment 图论->差分约束系统->无负权回路的有向图的最长路->Bellman-Ford1274 The Perfect Stall 图论->二分图的最大匹配1325 Machine Schedule 图论->二分图的最大匹配2239 Selecting Courses 图论->二分图的最大匹配2195 Going Home 图论->二分图的最大权匹配2400 Supervisor, Supervisee 图论->二分图的最大权匹配?1637 Sightseeing tour 图论->欧拉回路1383 Labyrinth 图论->树的最长路1094 Sorting It All Out 图论->拓扑排序1486 Sorting Slides 图论->拓扑排序1149 PIGS 图论->网络流2289 Jamie's Contact Groups 图论->网络流?1192 最优连通子集图论->无负权回路的有向图的最长路->BellmanFord 1364 King 图论->无负权回路的有向图的最长路->BellmanFord1985 Cow Marathon 图论->有向无环图的最长路1087 A Plug for UNIX 图论->最大流1273 Drainage Ditches 图论->最大流1459 Power Network 图论->最大流1632 Vase collection 图论->最大完全图2049 Finding Nemo 图论->最短路1251 Jungle Roads 图论->最小生成树2421 Constructing Roads 图论->最小生成树1026 Cipher 组合数学1095 Trees Made to Order 组合数学2346 Lucky tickets 组合数学1286 Necklace of Beads 组合数学->Polya定理2409 Let it Bead 组合数学->Polya定理1664 放苹果组合数学->递推。

陆家羲与组合设计大集

的 13 个 KT S( 15) . 在组合设计的术语中, 它被称为是 15 阶 Kir km an 三元系大集, 记作 LKT S( 15) . 该问题的答案直到 120 多年后的 1974 年由美国的 R. H . F . Dennist on 借助电子计算机才找到, 这
本身也说明了大集问题的难度. 以下是{ a, b} % Z13 上的一个 K T S( 15) , 它的元依次加上 0, 1,
2, . . . , 12 ( 模 13 取值, 但 a, b 保持不变) 将给出总共 13 个 KT S( 15) , 它们恰构成一个 LKT S( 15) .
S u n.
M o n.
T ue.
W ed.
T hu.
Fr i.
Sat.
{ 0, a, b} { 2, 8, b} { b, 11, 12} { 5, 7, b} { 4, 9, b} { 1, b, 10} { 3, 6, b}
{ 8, 9, 12} { 1, 6, a} { 4, a, 10} { 3, a, 12} { 2, 5, a} { 9, a, 11} { 7, 8, a}
{ 3, 7, 10} { 4, 7, 11} { 6, 7, 9} { 2, 9, 10} { 6, 8, 10} { 5, 6, 12} { 5, 10, 11}
彼此没有公共 3 子集的 v - 2 个 ST S( v ) , 但此时的阶数 v = 6t + 1 或 6t + 3, t 3 ( 这是 ST S( v )
存在的范围) . Sy lvest er 和 Cay ley 所提出的上述问题及一般的 LKT S( v) 存在性问题当时就引起了数学界的
广泛关注. 但 ST S( v ) & KT S( v ) 的可分解条件给原已很困难的大集问题进一步增添了难度. 在分析探索后, 人们退而求其次, 把精力首先集中到解决 L ST S( v) 的存在性问题上来. 1850 年, Cay ley

多项式时间判定素数

PRIMES is in P
Manindra Agrawal, Neeraj Kayal and Nitin Saxena∗ Department of Computer Science & Engineering Indian Institute of Technology Kanpur Kanpur-208016, INDIA August 6, 2002
2
Basic Idea and Approach
Our test is based on the following identity for prime numbers. This same identity was basis for a randomized polynomial-time algorithm in [AB99]: Identity Suppose that a is coprime to p. Then p is prime if and only if (x − a)p ≡ (xp − a)(mod p) (1)
∗ Emailຫໍສະໝຸດ addresses: manindra@cse.iitk.ac.in, kayaln@iitk.ac.in, nitinsa@cse.iitk.ac.in
1
The ultimate goal of this line of research is, of course, to obtain an unconditional deterministic polynomial-time algorithm for primality testing. Despite the impressive progress made in primality testing so far, this goal has remained elusive. In this paper, we achieve this. We give a deterministic, ˜ ((log n)12 ) time algorithm for testing if a number is prime. Heuristically, our algorithm does much O better: under a widely believed conjecture on the density of Sophie Germain primes (primes p such that ˜ ((log n)6 ) steps. The correctness proof of our algorithm 2p + 1 is also prime), the algorithm takes only O requires only simple tools of algebra (except for appealing to a sieve theory result on the density of primes p with p − 1 having a large prime factor). In contrast, the correctenss proofs of deterministic algorithms of [APR83, GK86, Atk86] are much more complex. In section 2, we summarize the basic idea behind our algorithm. In section 3, we state some preliminary theorems and ﬁx the notation used here. Thereafter, we state the algorithm in full detail and present the proof of correctness.

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DIAGONALCHECKER-JUMPINGANDEULERIANNUMBERSFORCOLOR-SIGNEDPERMUTATIONS

NIKLASERIKSEN,HENRIKERIKSSON,ANDKIMMOERIKSSONAbstract.Weintroducecolor-signedpermutationstoobtainaveryex-plicitcombinatorialinterpretationoftheq-EulerianidentitiesofBrentiandsomegeneralizations.Inparticular,weproveanidentityinvolvingthegoldenratio,whichallowsustocomputeupperboundsonhowhighacheckercanreachinaclassicalchecker-jumpingproblem,whentherulesarerelaxedtoallowalsodiagonaljumps.

1.IntroductionThispaperhastwothemes.First,wewillpresentanotionofcolor-signedpermutations,thatis,signedpermutationswheretheplussigncomesinpdiﬀerentcolorsandtheminussigncomesinqdiﬀerentcolors.Theseper-mutationsgiveaveryexplicitcombinatorialinterpretationoftheq-EulerianidentitiesofBrenti[2],andalsoprovidesomegeneralizationsofthem.Second,wewilltreatavariantofanoldchecker-jumpingproblem:Whenplayingcheckersonthesquaregrid,giventhatthelowerhalf-planeisfullofcheckersandtheupperhalf-planeisempty,howhighupintheupperhalf-planecanweplaceacheckerbyplayingthegame?Thisproblemwassolvedlongago(seeBerlekamp,ConwayandGuy’sentertainingbookoncombinatorialgames[1]).H.ErikssonandLindstr¨omgeneralizedchecker-jumpingtohigherdimensions[5].InhisM.Sc.thesis,Eriksen[4]changedtherulestoallowalsodiagonalchecker-jumpingmoves,therebyobtainingaharderproblemforwhichwewillherepresentanupperboundonhowhighthecheckercanbeplaced.Theconnectionbetweenthetwothemesisthefollowingidentity:Letnbeapositiveintegerandletσdenotethegoldenratio(√5−1)/2.Then

∞󰀆m=1m(2m+1)n−1σm+2=σ−2nn󰀆

k=1bnkσk,(1)

1991MathematicsSubjectClassiﬁcation.Primary:05A30;Secondary:05A99.Keywordsandphrases.Euleriannumbers,color-signedpermutations,checker-jumping,pagodafunction.N.E.:PartiallysupportedbytheSwedishNaturalScienceResearchCouncil.K.E.:PartiallysupportedbytheSwedishNaturalScienceResearchCouncilandtheSwedishFoundationforStrategicResearch.12NIKLASERIKSEN,HENRIKERIKSSON,ANDKIMMOERIKSSONwherethenumbersbnkaredeﬁnedbyb1,1=1,bnk=0fork≤0andfork>n,andtherecurrencerelation

bnk=(2(n−k)+1)bn−1,k−1+(2k+1)bn−1,k.Theﬁrstfewvaluesofbnkare:

k=1234n=1123139141427115491

Theidentity(1)isneededforcomputationoftheupperboundinthechecker-jumpingproblem.Zeilberger(personalcommunication)toldushow(1)couldbeprovedusinginduction.Ourownsearchforacombinatorialproofledustotheideaofcolor-signedpermutations.

2.PreliminariesonEulerianandq-EuleriannumbersLet[n]denotetheintegerset{1,...,n}.Signedpermutationson[n]deﬁneagroupdenotedbyBn.Thesubgroupconsistingofsignedpermutationswitheveryvalueplus-signedisisomorphictothesymmetricgroupSn.Wewilldescribeasignedpermutationπbyitssequenceπ1...πnofvalues.Twostatisticsonpermutationswillbeimportant:thenumberofdescentsandthenumberofminussigns.Adescentisapositioniwherethepermuta-tionvaluedecreases:πi−1>πi.Whencountingdescentswealwayshavetheconventionthatπ0=0.Letd(π)denotethenumberofdescentsofasignedpermutationπ.Thenumberofminussigns(resp.plussigns)inasignedpermutationπ∈BnisdenotedbyN(π)(resp.P(π)).Inotherwords,

N(π)def=#{i∈[n]:πi<0}andP(π)def=n−N(π).Thewell-knownEuleriannumbersAnkcountpermutationsinSnwithkdescents.Similarly,EuleriannumbersoftypeBarethenumbersBnk

ofsignedpermutationsinBnwithkdescents.Thesenumbersdeﬁnethe

Eulerianpolynomials:

An(x)def=n󰀆k=0AnkxkandBn(x)def=n󰀆k=0Bnkxk.ReﬁningalsobythestatisticN(π),Brenti[2]deﬁnedaq-analogtothesepolynomials,calledtheq-Eulerianpolynomial:

Bn(x;q)def=n󰀆k=0Bnk(q)xkdef=󰀆π∈BnqN(π)xd(π),wherethepolynomialsBnk(q)arecalledq-Euleriannumbers.Forq=1theyreducetotheEuleriannumbersoftypeB,andforq=0theyreducetotheordinaryEuleriannumbers.EULERIANNUMBERSFORCOLOR-SIGNEDPERMUTATIONS33.Color-signedpermutationsFromthedeﬁnitionoftheq-Euleriannumbers,itisclearthatifthevalueofqisanonnegativeinteger,thensoisBnk(q).Thereisasimplecombina-torialinterpretationofq,notmentionedbyBrenti:Ifwecoloreachminussignbyoneofqcolors,thenBnk(q)isthenumberofsuchq-colorsignedpermutationsinBnwithkdescents.(Coloringdoesnotaﬀectwhatismeantbyadescent.)Thisinterpretationallowsanobviousgeneralization,namely,coloringtheplussignsbyoneofpcolors.Deﬁnethepq-EuleriannumberBnk(p,q)asthenumberofpq-colorsignedpermutationsinBnwithkdescents.Aﬁrstobservationisthatn󰀆

k=0Bnk(p,q)=(p+q)nn!,

sincethereisachoiceamongpplussignsandqminussignsforeveryvalueofanunsignedpermutation.Thepq-EuleriannumbersareeasilyexpressedintermsofBrenti’sq-Euleriannumbers.

Lemma3.1.Thepq-Euleriannumbersarerelatedtotheq-Euleriannum-bersby

Bnk(p,q)=pnBnk(qp).

Proof.Sincethisisapolynomialidentity,itsuﬃcestoverifyitforallintegerspandqsuchthatqisdivisiblebyp.Givenonecolorforplussignsandqpcolorsforminussigns,spliteverycolorintopdiﬀerentsubcolors.Thisgives