On the maximum of r-Stirling numbers

The situation changes for Stirling numbers of the second kind. The sequence n n is strictly log-concave for all n [4,7,18,20] but no closed form is known k k=1 for the maximizing index Kn . Even we do not know whether it is unique or not. From 2 = 2 = 1, it is clear that K2 is not unique. For n ≥ 3 the uniqueness 1 2 of Kn was conjectured by Wegner [23]. This conjecture is supported by [4] where it is shown that Kn is unique for 3 < n < 106 . The papers [9,10,12–14,23] contain a number of estimations for Kn . The most exact (without any approximative term) was given by Wegner [23]: n log n − log log n

3 2 n
ζ (2) − ζ (3) Kn = log(n + 1) + γ − 1 + log(n + 1) + γ −
+
h log(n + 1) + γ −
3 2Βιβλιοθήκη 2,where [x] denotes the integer part of x, ζ is the Riemann zeta function, γ = 0.5772 . . . is the Euler-Mascheroni constant and −1.1 < h < 1.5. As Erd˝ os remarked, this can be simplified when n > 188: log n − 1 < Kn < [log n]. 2 2 (6)
As an equivalent definition, the numbers n are the coefficients of the k k=0 polynomial: n n k x = x(x + 1)(x + 2) · · · (x + n − 1). (2) k=0 k
n The Stirling number of the second kind, denoted by m , enumerates the number of partitions of a set with n elements consisting of m disjoint, nonempty sets. The following recurrence relation holds
(4)
An introduction on Stirling numbers can be found in [17]. A sequence a1 , a2 , . . . , an is said to be unimodal [27] if its members rise to a maximum and then decrease, that is, there exists an index k such that a1 ≤ a2 ≤ · · · ≤ ak , and ak ≥ ak+1 ≥ · · · ≥ an . The sequence a1 , a2 , . . . , an is called log-concave if a2 k ≥ ak+1 ak−1 (k = 2, . . . , n − 1), (5)
Abstract Determining the location of the maximum of Stirling numbers is a well developed area. In this paper we give same results for the so-called r-Stirling numbers which are natural generalizations of Stirling numbers. Key words: Stirling numbers, r-Stirling numbers, unimodality, log-concavity 1991 MSC: 11B73
n m n m n m
= 0,
r
n < r, n = r, n−1 m +
r
= δmr ,
r
(8)
= (n − 1)
r
n−1 , m−1 r
n > r.
A double generating function is given in [3]:
∞ n=0 n k=0
1 n + r k zn = . x n! (1 − z )r+x k+r
log n n log n e log n · · · ∼ . log n log n
2
(z > 3).
2
Notion of r-Stirling numbers
The r-Stirling numbers were introduced by Broder [3]. 3
n For positive integers n, m, r the r-Stirling numbers of the first kind m are r defined as the number of permutations of the set {1, 2, . . . , n} that have m cycles such that the elements 1, 2, . . . , r are in distinct cycles. The recurrence relation for r-Stirling numbers is the same as for ordinary Stirling numbers:
This immediately implies the strict version of (5). Newton’s inequality and (2) implies the strict log-concavity of n for all k k=1 n. According to the work of Hammersley [16] and Erd˝ os [8], much more is true. Namely, the index of the maximal Stirling number of the first kind is unique, i.e. for all n > 2 there is a unique index Kn such that n n n n n n < < ··· < < > > ··· > . 1 2 Kn − 1 Kn Kn + 1 n Moreover,
(9)
n For positive integers n, m, r the r-Stirling numbers of the second kind m r are defined as the number of partitions of the set {1, 2, . . . , n} that have m nonempty, disjoint subsets such that the elements 1, 2, . . . , r are in distinct subsets. They satisfy the recurrence relations
n
n = δ0n , 0
n n−1 n−1 =m + . m m m−1
(3)
1
Email address: imezo@math.klte.hu (Istv´ an Mez˝ o). URL: http://www.math.klte.hu/algebra/mezo.htm (Istv´ an Mez˝ o). Present address: University of Debrecen, H-4010, Debrecen, P.O. Box 12, Hungary
in the sense that their quotient tends to 1 as n tends to infinity. We remark that this approximation can be given using the result in [4]. It is shown that Kn ∈ { eW (n) − 1 , eW (n) − 1 }, where W (n) is the Lambert W function defined implicitly by the equation W (n)eW (n) = n. From the approximation [5, p. 349]: log log z log log z W (z ) = log z − log log z + +O log z log z we easily get eW (n) =
n m n m n m
= 0,
r
n < r, n = r, +
r
= δmr ,
r
(10)
=m
r
n−1 m
n−1 m−1
,
r
n > r.