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The Inequalities of Merris and Foregger for Permanents
- Source :
- Symmetry, Vol 13, Iss 1782, p 1782 (2021), Symmetry, Volume 13, Issue 10
- Publication Year :
- 2021
- Publisher :
- MDPI AG, 2021.
-
Abstract
- Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.
- Subjects :
- Doubly stochastic matrix
Conjecture
Physics and Astronomy (miscellaneous)
General Mathematics
MathematicsofComputing_GENERAL
permanent
Foregger’s inequality
Combinatorics
Matrix (mathematics)
TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES
Integer
Chemistry (miscellaneous)
Symmetric group
Linear algebra
Merris conjecture
Computer Science (miscellaneous)
QA1-939
Order (group theory)
Elementary symmetric polynomial
doubly stochastic matrices
Mathematics
Subjects
Details
- Language :
- English
- ISSN :
- 20738994
- Volume :
- 13
- Issue :
- 1782
- Database :
- OpenAIRE
- Journal :
- Symmetry
- Accession number :
- edsair.doi.dedup.....fccaae95ca9cd06253926e32e141a3aa