bibkey: chernfu2026signed
authors: Shane Chern, Shishuo Fu
year: 2026
title: "Signed counting of partition matrices"
doi: 10.48550/arXiv.2508.21318
url: https://arxiv.org/abs/2508.21318v2
claim: "Question 5.5 asks for a characterization of the fixed CDK image of improper partition matrices."
strata_touched:
  - D5/S3/Combinatorics/PartitionMatrix/ChernFuImage
license: citation-only
triage: anchor

Chern and Fu, signed counting of partition matrices

The source is arXiv:2508.21318v2, Section 5, Question 5.5:

Find a characterization (possibly in terms of pattern avoidance) of the inversion sequences produced by improper partition matrices under the action of the Claesson–Dukes–Kubitzke bijection.

Definition 1.1 specifies upper-triangular square set-entry matrices partitioning [n], with each row and column nonempty and weakly increasing column indices as labels increase. Following Theorem 1.2, the fixed map at label t sums the cardinalities of columns strictly before row_P(t). Definition 2.1 calls a same-column adjacent ascent or descent at i proper exactly when i has the parity of the least label in its column. Equal rows form neither row order.

The whole-alphabet gap condition in ChernFuImage is a characterization of this image. It uses A(e) = {e_1,…,e_n} and b_e(i) = max({0} union {a in A(e) : a < i}). For every 1 <= i < n, nonoccurrence of i and oddness of i - b_e(i) require e_i = e_(i+1). The criterion is about all inversion sequences and all native dimensions.

Version v1 Question 5.5 is a different bijection problem, numbered Question 5.7 in v2. That separate problem is not the assertion here.

The source has the arXiv nonexclusive distribution license. It is cited as mathematical literature; that license is not treated as a software license. No third-party implementation is copied.

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DOI: 10.48550/arXiv.2508.21318; versioned source: https://arxiv.org/abs/2508.21318v2.