The Lih–Wang permanent inequality toward the flat matrix
Let $\operatorname{per}$ be the permanent and $J_n$ the $n \times n$ matrix with every entry $1/n$, so that $\operatorname{per}(J_n) = n!/n^n$. Lih and Wang proved the inequality below for $n = 3$. For $n = 4$, Foregger proved it for $t_2 \le t \le 1$, where $t_2 \approx 0.6217$ is the least root of $106t^3 - 418t^2 + 465t - 153$.
Problem. Is $$\operatorname{per}\bigl(tJ_n + (1-t)A\bigr) \le t \operatorname{per}(J_n) + (1-t) \operatorname{per}(A)$$ for every $n \ge 3$, every doubly stochastic $A \in \mathbb{R}^{n \times n}$ and every $t \in [1/2, 1]$?
Source: Lih and Wang (1982); listed as Conjecture 34 in Cheon and Wanless (2005).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- CO
- Keywords
- doubly stochastic matrices, OpenProblemsInNLA, permanents
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{lih1982,
author = {Lih, Ko-Wei and Wang, Edward T. H.},
title = {A convexity inequality on the permanent of doubly stochastic matrices},
journal = {Congr. Numer.},
volume = {36},
year = {1982},
pages = {189--198},
}
@article{foregger1988,
author = {Foregger, Thomas H.},
title = {Permanents of convex combinations of doubly stochastic matrices},
journal = {Linear Multilinear Algebra},
volume = {23},
number = {1},
year = {1988},
pages = {79--90},
doi = {10.1080/03081088808817858},
}
@article{udayan2024,
author = {Udayan, Divya K. and Somasundaram, Kanagasabapathi},
title = {Lih {W}ang's and {D}ittert's conjectures on permanents},
journal = {Spec. Matrices},
volume = {12},
number = {1},
year = {2024},
pages = {Paper No. 20240006},
doi = {10.1515/spma-2024-0006},
eprint = {2312.00464},
archivePrefix = {arXiv},
}
@article{cheon2005,
author = {Cheon, Gi-Sang and Wanless, Ian M.},
title = {An update on {M}inc's survey of open problems involving permanents},
journal = {Linear Algebra Appl.},
volume = {403},
year = {2005},
pages = {314--342},
doi = {10.1016/j.laa.2005.02.030},
}
@misc{openproblemsinnla-mi17,
author = {Townsend, Alex},
title = {{MI-17}: The Lih–Wang permanent inequality toward the flat matrix},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-17/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00143, "The Lih–Wang permanent inequality toward the flat matrix", clopen.solutions.