{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T07:18:05Z","timestamp":1775805485933,"version":"3.50.1"},"reference-count":21,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,9,25]],"date-time":"2021-09-25T00:00:00Z","timestamp":1632528000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n\u00d7n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=\u2211\u03c3\u2208Sn\u220fi=1nai\u03c3(i). The Merris conjectured that for all n\u00d7n doubly stochastic matrices (denoted by \u03a9n), nperA\u2265min1\u2264i\u2264n\u2211j=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\u2212t)A)\u2264perA for 0\u2264t\u2264nn\u22121 and for all A\u2208\u03a9n, 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\u22654. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger\u2019s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n\u22655. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger\u2019s inequality for n=5 in [0.25, 0.6248]. Let \u03c3k(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen\u2013Dokovic conjecture), which states that if A\u2208\u03a9n,A\u2260Jn and k is an integer, 1\u2264k\u2264n, then \u03c3k(A)\u2265(n\u2212k+1)2nk\u03c3k\u22121(A). In this paper, we disprove the conjecture for n=k=4.<\/jats:p>","DOI":"10.3390\/sym13101782","type":"journal-article","created":{"date-parts":[[2021,9,27]],"date-time":"2021-09-27T23:08:31Z","timestamp":1632784111000},"page":"1782","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Inequalities of Merris and Foregger for Permanents"],"prefix":"10.3390","volume":"13","author":[{"given":"Divya K.","family":"Udayan","sequence":"first","affiliation":[{"name":"Department of Mathematics, Amrita School of Engineering, Amrita Vishwavidyapeetham, Coimbatore 641112, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2226-1845","authenticated-orcid":false,"given":"Kanagasabapathi","family":"Somasundaram","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Amrita School of Engineering, Amrita Vishwavidyapeetham, Coimbatore 641112, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,25]]},"reference":[{"key":"ref_1","first-page":"158","article-title":"Permanents","volume":"6","author":"Minc","year":"1978","journal-title":"Encycl. Math. Its Appl."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Lieb, E.H. (2002). Proofs of some conjectures on permanents. Inequalities, Springer.","DOI":"10.1007\/978-3-642-55925-9_11"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/0024-3795(86)90212-0","article-title":"A note on a conjecture on permanents","volume":"76","author":"Hwang","year":"1986","journal-title":"Linear Algebra Its Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1090\/S0002-9939-1981-0609645-5","article-title":"Monotonicity conjecture on permanents of doubly stochastic matrices","volume":"82","author":"Lih","year":"1981","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1016\/j.laa.2005.02.030","article-title":"An update on Minc\u2019s survey of open problems involving permanents","volume":"403","author":"Cheon","year":"2005","journal-title":"Linear Algebra Its Appl."},{"key":"ref_6","first-page":"305","article-title":"An update on a few permanent conjectures","volume":"4","author":"Zhang","year":"2016","journal-title":"Spec. Matrices"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1080\/00029890.1973.11993372","article-title":"The permanent of a doubly stochastic matrix","volume":"80","author":"Merris","year":"1973","journal-title":"Am. Math. Mon."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1080\/09720529.2015.1130813","article-title":"Some conjectures on permanents of doubly stochastic matrices","volume":"19","author":"Subramanian","year":"2016","journal-title":"J. Discret. Math. Sci. Cryptogr."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1080\/03081087908817268","article-title":"Remarks on a Conjecture of M. Marcus and H. Minc","volume":"7","author":"Foregger","year":"1979","journal-title":"Linear Multilinear Algebra"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1080\/03081088808817858","article-title":"Permanents of convex combinations of doubly stochastic matrices","volume":"23","author":"Foregger","year":"1988","journal-title":"Linear Multilinear Algebra"},{"key":"ref_11","unstructured":"Holens, F. (1964). Two Aspects of Doubly Stochastic Matrices: Permutation Matrices and the Minimum Permanent Function. [Ph.D. Thesis, University of Manitoba]."},{"key":"ref_12","first-page":"566","article-title":"On a conjecture by van der Waerden","volume":"19","author":"Djokovic","year":"1967","journal-title":"Mat. Vesnik"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/0024-3795(86)90296-X","article-title":"The monotonicity of and the \u0110okovi\u0107 conjectures on permanents of doubly stochastic matrices","volume":"79","author":"Hwang","year":"1986","journal-title":"Linear Algebra Its Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/S0024-3795(98)10177-5","article-title":"The Holen-Dokovic conjecture on permanents fails","volume":"286","author":"Wanless","year":"1999","journal-title":"Linear Algebra Its Appl."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Seneta, E. (1981). Non-Negative Matrices and Markov Chains, Springer.","DOI":"10.1007\/0-387-32792-4"},{"key":"ref_16","first-page":"189","article-title":"A convexity inequality on the permanent of doubly stochastic matrices","volume":"36","author":"Lih","year":"1982","journal-title":"Congr. Numer."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1017\/S0305004100041219","article-title":"On a conjecture of B. L. Van der Waerden","volume":"63","author":"Marcus","year":"1967","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1080\/03081087708817189","article-title":"On a conjecture of M. Marcus and H. Minc","volume":"5","author":"Wang","year":"1977","journal-title":"Linear Multilinear Algebra"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1016\/0024-3795(69)90033-0","article-title":"Remarks on the van der Waerden conjecture II","volume":"2","author":"Eberlein","year":"1969","journal-title":"Linear Algebra Its Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"386","DOI":"10.1016\/S0021-9800(68)80015-8","article-title":"Some remarks on the van der Waerden conjecture","volume":"5","author":"Eberlein","year":"1968","journal-title":"J. Comb. 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