Signed counting of partition matrices

2025/08/29 by Chern, Shane, Fu, Shishuo
#05A05 #05A15 #05A19 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.21318

Abstract

We prove that the signed counting (with respect to the parity of the ``inv'' statistic) of partition matrices equals the cardinality of a subclass of inversion sequences. In the course of establishing this result, we introduce an interesting class of partition matrices called improper partition matrices. We further show that a subset of improper partition matrices is equinumerous with the set of Motzkin paths. Such an equidistribution is established both analytically and bijectively.

Citations

• Equidistributed statistics on Fishburn matrices and permutations
• Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations
• On q-Series Identities Related to Interval Orders
• (2+2)-free posets, ascent sequences and pattern avoiding permutations
• Symmetry classes of alternating-sign matrices under one roof
• Permutations with restricted patterns and Dyck paths

Related

• Weighted counting of inversions on alternating sign matrices
• The part-frequency matrices of a partition
• The relation between alternating sign matrices and descending plane partitions: n+3 pairs of equivalent statistics
• Signed Counting on Restricted Partitions and Combinatorial Proofs of Three Identities
• A bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruplet of statistics considered by Behrend, Di Francesco and Zinn--Justin