On a conjecture concerning the Bruhat order
Let R and S be two sequences of positive integers in nonincreasing order having the same sum. Let A(R,S) be the class of all (0,1)-matrices with row sum vector R and column sum vector S. If A(R,S) is nonempty, an inversion in A∈A(R,S) consists of two entries of A equal to 1, one of them is located t...
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| Veröffentlicht in: | Linear algebra and its applications 2020-09, Vol.600, p.82-95 |
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| creator | Fernandes, Rosário da Cruz, Henrique F. Salomão, Domingos |
| description | Let R and S be two sequences of positive integers in nonincreasing order having the same sum. Let A(R,S) be the class of all (0,1)-matrices with row sum vector R and column sum vector S. If A(R,S) is nonempty, an inversion in A∈A(R,S) consists of two entries of A equal to 1, one of them is located to the top-right of the other. Let ν(A) be the total number of inversions in A. The Bruhat order is a partial order defined on A(R,S) and denoted by ⪯B. In this paper, we prove the conjecture:•“If A,C∈A(R,S), A≠C and A⪯BC then ν(A) |
| doi_str_mv | 10.1016/j.laa.2020.04.015 |
| format | Article |
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Let A(R,S) be the class of all (0,1)-matrices with row sum vector R and column sum vector S. If A(R,S) is nonempty, an inversion in A∈A(R,S) consists of two entries of A equal to 1, one of them is located to the top-right of the other. Let ν(A) be the total number of inversions in A. The Bruhat order is a partial order defined on A(R,S) and denoted by ⪯B. In this paper, we prove the conjecture:•“If A,C∈A(R,S), A≠C and A⪯BC then ν(A)<ν(C)”.</description><identifier>ISSN: 0024-3795</identifier><identifier>EISSN: 1873-1856</identifier><identifier>DOI: 10.1016/j.laa.2020.04.015</identifier><language>eng</language><publisher>Amsterdam: Elsevier Inc</publisher><subject>[formula omitted]-Matrices ; Bruhat order ; Inversions ; Linear algebra ; Mathematical analysis ; Matrix algebra ; Matrix methods ; Sequences</subject><ispartof>Linear algebra and its applications, 2020-09, Vol.600, p.82-95</ispartof><rights>2020 Elsevier Inc.</rights><rights>Copyright American Elsevier Company, Inc. 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Let A(R,S) be the class of all (0,1)-matrices with row sum vector R and column sum vector S. If A(R,S) is nonempty, an inversion in A∈A(R,S) consists of two entries of A equal to 1, one of them is located to the top-right of the other. Let ν(A) be the total number of inversions in A. The Bruhat order is a partial order defined on A(R,S) and denoted by ⪯B. In this paper, we prove the conjecture:•“If A,C∈A(R,S), A≠C and A⪯BC then ν(A)<ν(C)”.</description><subject>[formula omitted]-Matrices</subject><subject>Bruhat order</subject><subject>Inversions</subject><subject>Linear algebra</subject><subject>Mathematical analysis</subject><subject>Matrix algebra</subject><subject>Matrix methods</subject><subject>Sequences</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNp9kMtOwzAQRS0EEqXwAewisUQJ42dssYKqPKRK3cDacuwpTVSSYidI_D2uyprVzOKeeRxCrilUFKi666qdcxUDBhWICqg8ITOqa15SLdUpmQEwUfLayHNykVIHAKIGNiO3675whR_6Dv04RTy0HmPf9h_FuMXiMU5bNxZDDBgvydnG7RJe_dU5eX9avi1eytX6-XXxsCo9V3os68b7DUAwNTohvZEopDHImVOsMY1ywTgZXI5R2gSttGPGGeAZYJo1nM_JzXHuPg5fE6bRdsMU-7zSMiGEqrkGyCl6TPk4pBRxY_ex_XTxx1KwBye2s9mJPTixIGx2kpn7I4P5_O8Wo02-xfxwaGP-34ah_Yf-BQASZ9w</recordid><startdate>20200901</startdate><enddate>20200901</enddate><creator>Fernandes, Rosário</creator><creator>da Cruz, Henrique F.</creator><creator>Salomão, Domingos</creator><general>Elsevier Inc</general><general>American Elsevier Company, Inc</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0003-2695-9079</orcidid></search><sort><creationdate>20200901</creationdate><title>On a conjecture concerning the Bruhat order</title><author>Fernandes, Rosário ; da Cruz, Henrique F. ; Salomão, Domingos</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-7bccf00d97ea45c95e4599e32a62b9b6ad9a5da7bc11bd868a29a903d97282b33</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>[formula omitted]-Matrices</topic><topic>Bruhat order</topic><topic>Inversions</topic><topic>Linear algebra</topic><topic>Mathematical analysis</topic><topic>Matrix algebra</topic><topic>Matrix methods</topic><topic>Sequences</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Fernandes, Rosário</creatorcontrib><creatorcontrib>da Cruz, Henrique F.</creatorcontrib><creatorcontrib>Salomão, Domingos</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Linear algebra and its applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Fernandes, Rosário</au><au>da Cruz, Henrique F.</au><au>Salomão, Domingos</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On a conjecture concerning the Bruhat order</atitle><jtitle>Linear algebra and its applications</jtitle><date>2020-09-01</date><risdate>2020</risdate><volume>600</volume><spage>82</spage><epage>95</epage><pages>82-95</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>Let R and S be two sequences of positive integers in nonincreasing order having the same sum. Let A(R,S) be the class of all (0,1)-matrices with row sum vector R and column sum vector S. If A(R,S) is nonempty, an inversion in A∈A(R,S) consists of two entries of A equal to 1, one of them is located to the top-right of the other. Let ν(A) be the total number of inversions in A. The Bruhat order is a partial order defined on A(R,S) and denoted by ⪯B. In this paper, we prove the conjecture:•“If A,C∈A(R,S), A≠C and A⪯BC then ν(A)<ν(C)”.</abstract><cop>Amsterdam</cop><pub>Elsevier Inc</pub><doi>10.1016/j.laa.2020.04.015</doi><tpages>14</tpages><orcidid>https://orcid.org/0000-0003-2695-9079</orcidid><oa>free_for_read</oa></addata></record> |
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| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | [formula omitted]-Matrices Bruhat order Inversions Linear algebra Mathematical analysis Matrix algebra Matrix methods Sequences |
| title | On a conjecture concerning the Bruhat order |
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