Research Preprints · Combinatorics

A Koteljanskii inequality for permanents

Suvrit Sra · TU Munich, Department of Mathematics, School of CIT

Version v1 · released 2 October 2026 · 9 pages

PDFarXiv:2609.39979v1GitHubTimestamp

Abstract

We prove a permanental analogue of Koteljanskii's inequality. If $A$ is an inverse $M$-matrix that becomes symmetric after a positive diagonal similarity, then $\mathrm{per}(A_{S\cup T})\,\mathrm{per}(A_{S\cap T})\ge\mathrm{per}(A_S)\,\mathrm{per}(A_T)$ for all $S,T\subseteq[n]$, where $A_S$ is the principal submatrix indexed by $S$. The proof expresses permanents as moments of a complex Gaussian vector and uses Ginibre's correlation inequality. Finally, an explicit counterexample shows that symmetry cannot be dropped.

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Cite

@misc{sra2026perm-m-matrix-v1,
  author        = {Suvrit Sra},
  title         = {{A Koteljanskii inequality for permanents}},
  year          = {2026},
  month         = sep,
  howpublished  = {\url{https://arxiv.org/abs/2609.39979v1}},
  eprint        = {2609.39979v1},
  archiveprefix = {arXiv},
  note          = {Also released as perm-m-matrix v1 (2 October 2026) at \url{https://github.com/suvrit/preprints/tree/main/preprints/perm-m-matrix/v1}},
}

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