Tamás Lengyel

On a Recurrence involving Stirling Numbers

We count the number Z ( n ) of (not necessarily maximal) chains from 0 to 1 in the partition lattice of an n -set. This function satisfies the recurrence Z ( n ) = ∑ k − 1 n − 1 S ( n , k ) Z ( k ) where S ( n , k ) denotes the Stirling numbers of the second kind. We find the asymptotic order of magnitude of Z ( n ).

A combinatorial identity and the World Series

In this note the author gives a simple probabilistic proof of a combinatorial identity by calculating the winning probability in the World Series. The winning probabilities and the expected length of the championship series are given by the applications of the identity and its generalization.

The Sprague-Grundy function of the real game Euclid

The game Euclid, introduced and named by Cole and Davie, is played with a pair of nonnegative integers. The two players move alternately, each subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who reduces one of the integers to zero wins. Unfortunately, the name Euclid has also been used for a subtle variation of this game due to Grossman in which the game stops when the two entries are equal. For that game, Straffin showed that the losing positions (a,b) with a

The game of 3-Euclid

In this paper we study 3-Euclid, a modification of the game Euclid to three dimensions. In 3-Euclid, a position is a triplet of positive integers, written as (a,b,c). A legal move is to replace the current position with one in which any integer has been reduced by an integral multiple of some other integer. The only restriction on this subtraction is that the result must stay positive. We solve the game for some special cases and prove two theorems which give some properties of 3-Euclids Sprague-Grundy function. They provide a structural description of all positions of Sprague-Grundy value g with two numbers fixed. We state a theorem which establishes a periodicity in the P positions (i.e., those of Sprague-Grundy value g=0), and extend some results to the misere version.

Alternative Proofs on the 2-adic Order of Stirling Numbers of the Second Kind

Abstract An interesting 2-adic property of the Stirling numbers of the second kind S(n, k) was conjectured by the author in 1994 and proved by De Wannemacker in 2005: ν 2(S(2 n , k)) = d 2(k) – 1, 1 ≤ k ≤ 2 n . It was later generalized to ν 2(S(c2 n , k)) = d 2(k) – 1, 1 ≤ k ≤ 2 n , c ≥ 1 by the author in 2009. Here we provide full and two partial alternative proofs of the generalized version. The proofs are based on non-standard recurrence relations for S(n, k) in the second parameter and congruential identities.

On some properties of the series k=0 ∞ k n x k and the Stirling numbers of the second kind

Abstract We partially characterize the rational numbers x and integers n ⩾ 0 for which the sum ∑k=0∞ knxk assumes integers. We prove that if ∑k=0∞ knxk is an integer for x = 1 − a/b with a, b > 0 integers and gcd(a,b) = 1, then a = 1 or 2. Partial results and conjectures are given which indicate for which b and n it is an integer if a = 2. The proof is based on lower bounds on the multiplicities of factors of the Stirling number of the second kind, S(n,k). More specifically, we obtain ν a (n−k)! ⩾ν a (n!)−k+1 for all integers k, 2 ⩽ k ⩽ n, and a ⩾ 3, provided a is odd or divisible by 4, where va(m) denotes the exponent of the highest power of a which divides m, for m and a > 1 integers. New identities are also derived for the Stirling numbers, e.g., we show that ∑k=02nk! S(2n, k) − 1 2 k =0 , for all integers n ⩾ 1.

A CONVERGENCE CRITERION FOR RECURRENT SEQUENCES WITH APPLICATION TO THE PARTITION LATTICE

We prove a fairly general convergence criterion for sequences satisfying a linear recurrence (defined by an infinite triangular matrix). We prove that every sequence of positive numbers satisfying a nearly convex linear recurrence with finite retardation and active predecessors converges to a positive limit. Informally, near convexity means the coefficients axe nonnegative and the sum of coefficients in each equation is approximately 1; finite retardation means low order terms have little weight; and active predecessors mean that the immediate predecessor carries a weight greater than a fixed positive constant. We present an application to the asymptotic number of not necessarily maximal chains in the partition lattice. The coefficients of the corresponding recurrence are the Stirling numbers of the second kind. AMS 1980 classification numbers: 40A05, 11B37, 05A15, 06C10, 11B73

EXACT p-ADIC ORDERS FOR DIFFERENCES OF MOTZKIN NUMBERS

For any prime p, we establish congruences modulo pn+1 for the difference of the pn+1th and pnth Motzkin numbers and determine the p-adic order of the difference. The results confirm recent conjectures on the order. The applied techniques involve the use of congruences for the differences of certain Catalan numbers and binomial coefficients, congruential identities for sums of Catalan numbers, central binomial and trinomial coefficients, infinite incongruent disjoint covering systems and the solution of congruential recurrences.

On the rate of p-adic convergence of sums of powers of binomial coefficients

Let m ≥ 1 be an integer and p be an odd prime. We study sums and lacunary sums of mth powers of binomial coefficients from the point of view of arithmetic properties. We develop new congruences and prove the p-adic convergence of some subsequences and that in every step we gain at least one or three more p-adic digits of the limit if m = 1 or m ≥ 2, respectively. These gains are exact under some explicitly given conditions. The main tools are congruential and divisibility properties of the binomial coefficients and multiple and alternating harmonic sums.

On Some 2-Adic Properties of a Recurrence Involving Stirling Numbers ∗

We analyze some 2-adic properties of the sequence defined by the recurrence Z(1) = 1; Z(n) = Σk=1n−1S(n, k)Z(k), n ≥ 2, which counts the number of ultradissimilarity relations, i.e., ultrametrics on an n-set. We prove the 2-adic growth property ν2(Z(n)) ≥ ⌈log2n⌉ −1 and present conjectures on the exact values.