From Unit Interval Orders to Totally Nonnegative Matrices: Permanental Inequalities and Bruhat-Increasing Bijections
Abstract
Given an n× n matrix A=(ai,j) and subsets I,J of [1,n]:={1,dotsc,n}, let AI,J=(ai,j)i∈ I,j∈ J denote the (I,J)-submatrix of A. We prove that, for every totally nonnegative matrix A and every h=1,dotsc,n−1, perm(A[1,h],[1,h]) perm(A[h+1,n],[h+1,n]) ≥ perm(A[1,n]∩ 2ℤ,[1,n]∩ 2ℤ) perm(A[1,n]smallsetminus 2ℤ,[1,n]smallsetminus 2ℤ). labeleq:absmain In particular, these inequalities hold for the $0−1$ totally nonnegative matrices arising as antiadjacency matrices of unit interval orders. Our proof is based on explicit Bruhat-order comparisons between two distinguished families of permutations. Let An be the Young subgroup S1,…,⌊n/2⌋ × S⌊n/2⌋+1,…,n and let Bn be the set of parity-alternating permutations {w=w1⋯ wn∈Sn | i and wi have the same parity}. For the balanced case h=⌊n/2⌋, we construct a bijection fn:An→Bn satisfying w≤ fn(w) in Bruhat order. We then extend this construction to arbitrary h by means of maps gn,h satisfying the corresponding Bruhat-order comparison. These maps yield the inequalities for all totally nonnegative matrices.
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BibTeX
@misc{pan2026from,
title = {{From Unit Interval Orders to Totally Nonnegative Matrices: Permanental Inequalities and Bruhat-Increasing Bijections}},
author = {Sihong Pan and Mark Skandera and Jiayuan Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04809},
eprint = {2610.04809},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04809},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Unit Interval Orders to Totally Nonnegative Matrices: Permanental Inequalities and Bruhat-Increasing Bijections
AU - Sihong Pan
AU - Mark Skandera
AU - Jiayuan Wang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.04809
UR - https://arxiv.org/abs/2610.04809
AB - Given an n× n matrix A=(a_i,j) and subsets I,J of [1,n]:={1,dotsc,n}, let A_I,J=(a_i,j)_i∈ I,j∈ J denote the (I,J)-submatrix of A. We prove that, for every totally nonnegative matrix A and every h=1,dotsc,n−1, perm(A_[1,h],[1,h]) perm(A_[h+1,n],[h+1,n]) ≥ perm(A_[1,n]∩ 2ℤ,[1,n]∩ 2ℤ) perm(A_[1,n]smallsetminus 2ℤ,[1,n]smallsetminus 2ℤ). labeleq:absmain In particular, these inequalities hold for the $0−1$ totally nonnegative matrices arising as antiadjacency matrices of unit interval orders. Our proof is based on explicit Bruhat-order comparisons between two distinguished families of permutations. Let A_n be the Young subgroup S_1,…,⌊n/2⌋ × S_⌊n/2⌋+1,…,n and let B_n be the set of parity-alternating permutations {w=w_1⋯ w_n∈S_n | i and w_i have the same parity}. For the balanced case h=⌊n/2⌋, we construct a bijection f_n:A_n→B_n satisfying w≤ f_n(w) in Bruhat order. We then extend this construction to arbitrary h by means of maps g_n,h satisfying the corresponding Bruhat-order comparison. These maps yield the inequalities for all totally nonnegative matrices.
ER -