Refined Counting of Core Partitions into $d$-Distinct Parts

Hannah Burson,Simone Sisneros-Thiry,Armin Straub

doi:10.37236/9665

Abstract

Using a combinatorial bijection with certain abaci diagrams, Nath and Sellers have enumerated $(s,ms\pm 1)$-core partitions into distinct parts. We generalize their result in several directions by including the number of parts of these partitions, by considering $d$-distinct partitions, and by allowing more general $(s,ms\pm r)$-core partitions. As an application of our approach, we obtain the average and maximum number of parts of these core partitions.

Highlights

If λ1 λ2 · · · λr 1, where the parts λi are integers, λ = (λ1, λ2, . . . , λr) is an partition of |λ| = λ1 + · · · + λr
The Young diagram of a partition λ is a left-justified array of square cells, where the first row contains λ1 cells, the second row λ2 cells, and so on
In the case 1 r d, applying Lemma 5 with ms in place of s, we find that every (s, ms + r)-core partition into d-distinct parts has maximum hook length < ms + r

Summary

Introduction

Nath and Sellers [13] proved (the case of (s, ms + 1)-core partitions of) Theorem 1 combinatorially by viewing the partitions as certain abaci In light of Lemma 6, we prove Theorem 2 as a special case of our Theorems 12 and 14 (as well as Lemma 15, in the case r = −1 and s = 2d + 2, and Lemma 17 in the case r = −1 and d = 1) on certain abaci It follows from Theorem 12 that the condition r d can be dropped if is modified to be the generating polynomial of parts of s-core partitions into d-distinct parts with largest hook of length less than ms + r.

Preliminaries and s-core abaci

Maximum number of parts and partitions with maximal initial gaps

Average number of parts

Conclusion