Related papers: Wilf conjecture

Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not…

In a Note in this Monthly, Klazar raised the question of whether the alternating sum of the Stirling numbers of the second kind $B^\pm(n)=\sum_{k=0}^n(-1)^kS(n,k)$ is ever zero for $n\neq 2$. In this article, we present an exposition of the…

Let $n,k,a$ and $c$ be positive integers and $b$ be a nonnegative integer. Let $\nu_2(k)$ and $s_2(k)$ be the 2-adic valuation of $k$ and the sum of binary digits of $k$, respectively. Let $S(n,k)$ be the Stirling number of the second kind.…

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna…

A formula on Stirling numbers of the second kind $S(n, k)$ is proved. As a corollary, for odd $n$ and even $k$, it is shown that $k!S(n, k)$ is a positive multiple of the greatest common divisor of $j!S(n, j)$ for $k+1\leq j\leq n$. Also,…

For positive integers m and n, denote S(m,n) as the associated Stirling number of the second kind and let z be a complex variable. In this paper, we introduce the Stirling functions S(m,n,z) which satisfy S(m,n,z) = S(m,n) for any z which…

Let K_n denote the smaller mode of the nth row of Stirling numbers of the second kind S(n, k). Using a probablistic argument, it is shown that for all n>=2, [exp(w(n))]-2<=K_n<=[exp(w(n))]+1, where [x] denotes the integer part of x, and…

Let $n$ and $k$ be positive integers. We denote by $v_2(n)$ the 2-adic valuation of $n$. The Stirling numbers of the first kind, denoted by $s(n,k)$, counts the number of permutations of $n$ elements with $k$ disjoint cycles. In recent…

We generalize the Stirling numbers of the first kind $s(a,k)$ to the case where $a$ may be an arbitrary real number. In particular, we study the case in which $a$ is an integer. There, we discover new combinatorial properties held by the…

Let $m, n, k$ and $c$ be positive integers. Let $\nu_2(k)$ be the 2-adic valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of…

We study the series s(n,x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson's theorem we show that the integer part of s(n,x) for x = Pi/2 is the number of primes…

We study Wilf's conjecture for numerical semigroups $S$ such that the second least generator $a_2$ of $S$ satisfies $a_2>\frac{c(S)+\mu(S)}{3}$, where $c(S)$ is the conductor and $\mu(S)$ the multiplicity of $S$. In particular, we show that…

It is known that the Scholz conjecture on addition chains is true for all integers $n$ with $\ell(2n) = \ell(n)+1$. There exists infinitely many integers with $\ell(2n) \leq \ell(n)$ and we don't know if the conjecture still holds for them.…

For the Stirling numbers of the second kind $S(n,k)$ and the ordered Bell numbers $B(n)$, we prove the identity $\sum_{k=1}^{n/2} S(n,2k)(2k-1)! = B(n-1)$. An analogous identity holds for the sum over odd $k$'s.

Given a finite nonempty sequence of integers S, by grouping adjacent terms it is always possible to write it, possibly in many ways, as S = X Y^k, where X and Y are sequences and Y is nonempty. Choose the version which maximizes the value…

We analyze congruence classes of $S(n,k)$, the Stirling numbers of the second kind, modulo powers of 2. This analysis provides insight into a conjecture posed by Amdeberhan, Manna and Moll, which those authors established for $k\le5$. We…

Using the theory of spherical varieties and especially Frobenius splitting results for symmetric varieties, we give a type independent very short proof of Wahl's conjecture for cominuscule homogeneous spaces for all primes different from 2.

In this note, we show that $S(n,r):=\sum_{k=0}^{n} \binom{n}{k}\frac{k}{k+r}$ is not an integer for any positive integer $n$ and $r\in \{1,2,3,4,5,6\}$ and for $n\le r-1$. This gives a partial answer to a conjecture of [3].

We show the Williams Conjecture is false for irreducible shifts of finite type by examining relative sign-gyration numbers of conjugacies between shifts with no points of period one or two.

The partial Stirling numbers T_n(k) used here are defined as the sum over odd values of i of (n choose i) i^k. Their 2-exponents nu(T_n(k)) are important in algebraic topology. We provide many specific results, applying to all values of n,…