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.
We consider two families of plane partitions: totally symmetric self-complementary plane partitions (TSS-CPPs) and cyclically symmetric transpose complement plane partitions (CSTCPPs). If T (n) and C(n) are the numbers of such plane partitions in a 2n × 2n × 2n box, then ord 2 (T (n)) = ord 2 (C(n)) for all n ≥ 1. We also discuss various consequences, along… (More)
In this note, we consider arithmetic properties of the function K(n) = (2n)!(2n + 2)! (n − 1)!(n + 1)! 2 (n + 2)! which counts the number of two–legged knot diagrams with one self– intersection and n − 1 tangencies. This function recently arose in a paper by Jacobsen and Zinn–Justin on the enumeration of knots via a transfer matrix approach. Using… (More)