The $(s,s+d,\dots ,s+pd)$-core partitions and the rational Motzkin paths

Abstract

In this paper, we propose an $(s+d,d)$-abacus for $(s,s+d,\dots ,s+pd)$-core partitions and establish a bijection between the $(s,s+d,\dots ,s+pd)$-core partitions and the rational Motzkin paths of type $(s+d,-d)$. This result not only gives a lattice path interpretation of the $(s,s+d,\dots ,s+pd)$-core partitions but also counts them with a closed formula. Also we enumerate $(s,s+1,\dots ,s+p)$-core partitions with $k$ corners and self-conjugate $(s,s+1,\dots ,s+p)$-core partitions.

11 pages

References in corpus (1)

• Counting self-conjugate (s,s+1,s+2)-core partitions