The paper gives formulas for the number of compositions of n whose ribbon number is congruent to i modulo p, including explicit cases n=mp^d and sums of distinct powers of p, with extensions to Coxeter groups of types B and D.
Enumeration of Odd-Dimensional Partitions modulo 4
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abstract
The number of standard Young tableaux of shape a partition $\lambda$ is called the dimension of the partition and is denoted by $f^{\lambda}$. Partitions with odd dimensions were enumerated by McKay and were further characterized by Macdonald. Let $a_i(n)$ be the number of partitions of $n$ with dimension congruent to $i$ modulo 4. In this paper, we refine Macdonald's and McKay's results by computing $a_1(n)$ and $a_3(n)$ when $n$ has no consecutive 1s in its binary expansion or when the sum of binary digits of $n$ is 2.
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Dimensions of compositions modulo a prime
The paper gives formulas for the number of compositions of n whose ribbon number is congruent to i modulo p, including explicit cases n=mp^d and sums of distinct powers of p, with extensions to Coxeter groups of types B and D.