A partition n = p 1 + p 2 + · · · + p k with 1 ≤ p 1 ≤ p 2 ≤ · · · ≤ p k is called non-squashing if p 1 + · · · + p j ≤ p j+1 for 1 ≤ j ≤ k − 1. Hirschhorn and Sellers showed that the number of non-squashing partitions of n is equal to the number of binary partitions of n. Here we exhibit an explicit bijection between the two families, and determine the… (More)
γνωσι πoλλoυς αι µπνoιης The goal of this paper is to prove two new congruences involving 4–cores using elementary techniques; namely, if a 4 (n) denotes the number of 4–cores of n, then a 4 (9n+2) ≡ 0 (mod 2) and a 4 (9n + 8) ≡ 0 (mod 4).
Recently, Sloane and Sellers solved a certain box stacking problem related to non– squashing partitions. These are defined as partitions n = p 1 + p 2 + · · · + p k with 1 ≤ p 1 ≤ p 2 ≤ · · · ≤ p k wherein p 1 + · · · + p j ≤ p j+1 for 1 ≤ j ≤ k − 1. Sloane has also hinted at a generalized box stacking problem which is closely related to generalized… (More)
Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials D n (x, a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are… (More)
Let A(n) denote the number of n × n alternating sign matrices and J m the m th Jacobsthal number. It is known that A(n) = n−1 ℓ=0 (3ℓ + 1)! (n + ℓ)!. The values of A(n) are in general highly composite. The goal of this paper is to prove that A(n) is odd if and only if n is a Jacobsthal number, thus showing that A(n) is odd infinitely often.
For a fixed integer m ≥ 2, we say that a partition n = p 1 + p 2 + · · · + p k of a natural number n is m-non-squashing if p 1 ≥ 1 and (m − 1)(p 1 + · · · + p j−1) ≤ p j for 2 ≤ j ≤ k. In this paper we give a new bijective proof that the number of m-non-squashing partitions of n is equal to the number of m-ary partitions of n. Moreover, we prove a similar… (More)
×ØÖÖØº We give a variety of results involving s(n), the number of representations of n as a sum of three squares. Using elementary techniques we prove that if 9 ¹ n, s(9 λ n) = 3 λ+1 − 1 2 − −n 3 3 λ − 1 2 s(n); via the theory of modular forms of half integer weight we prove the corresponding result with 3 replaced by p, an odd prime. This leads to a… (More)
In a recent work, M. Covington discusses the enumeration of two different sets of alignments of two strings of symbols using elementary combinatorial techniques. He defines two functions a(m, n) and A(m, n) to count the number of two–string alignments in his " small " and " middle " sets of alignments (respectively). He provides a recurrence for each of… (More)
In 1962, S. L. Hakimi proved necessary and sufficient conditions for a given sequence of positive integers d1, d2,. .. , dn to be the degree sequence of a non–separable graph or that of a connected graph. Our goal in this note is to utilize these results to prove closed formulas for the functions dns(2m) and dc(2m), the number of degree sequences with… (More)