bibkey: barrese2023levels
authors: Kenneth Barrese
year: 2023
title: "Weighted File Placements on Singleton Boards"
doi: 10.48550/arXiv.2312.10855
url: https://arxiv.org/abs/2312.10855v1
claim: "Sections 2.2 and 5 ask for the number of all Ferrers boards with prescribed ordered level numbers in the fixed n-column, n-level ambient rectangle."
strata_touched:
  - D5/S3/Combinatorics/Partitions/FerrersLevelCounting
license: citation-only
triage: anchor

Prescribed ordered level numbers

Verified locator

DOI: 10.48550/arXiv.2312.10855. URL: https://arxiv.org/abs/2312.10855v1. The verified scope is Section 2.2, the ordered level numbers in Theorem 4 and the following open counting question, and the first open question in Section 5. It concerns all Ferrers boards in the fixed n-column, mn-row ambient rectangle, with levels indexed from the top for the prescribed ordered data, including zero columns and empty levels.

The separate 2015 source is Barrese, Loehr, Remmel and Sagan, Bijections on m-level Rook Placements, https://arxiv.org/abs/1508.05944v1, Definition 4 in Section 2.2 and the counting question in Section 6.2. Its l-operator fiber ranges over all normalized widths for a prescribed singleton target; zero-padding normalization and the separately declared empty-board extension retain the boundaries stated below.

Source scope

In Section 2.2 an ambient board has n columns and mn rows, divided into n levels of m rows, counted from the bottom. The ordered level numbers in Theorem 4 and the following open question are instead indexed from the top. The unknown boards in this question are arbitrary Ferrers boards; only other results in the paper require singleton boards. Leading zero columns and empty top levels remain part of this fixed ambient description. Section 5 repeats the counting question as its first open question.

The related question in Barrese, Loehr, Remmel and Sagan, Bijections on m-level Rook Placements, arXiv:1508.05944v1, Section 6.2, fixes a singleton board C and asks for all Ferrers boards B with l(B)=C. Definition 4 in Section 2.2 defines l(B) by reversing the bottom-to-top list of nonempty level totals. That formulation has no prescribed input width. The paper uses left zero padding of column tuples elsewhere; normalized shapes under that convention must be distinguished from literal finite subsets of the positive absolute coordinate grid. The empty-board extension of Definition 4 is a separate convention because its largest nonempty level is undefined.

Neither question asks here for the size of a whole m-level rook equivalence class. Gaussian polynomials used to count row configurations are ordinary auxiliary generating polynomials, unrelated to the file-placement weights of the 2023 paper.