Schur-Positivity of Short Chords in Matchings
We prove that the set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. Two proofs are presented. The first proof applies a new combinatorial criterion for Schur-positivity, while the second is bijective. The coefficients in the Schur e...
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| creator | Marmor, Avichai |
| description | We prove that the set of matchings with a fixed number of unmatched vertices
is Schur-positive with respect to the set of short chords. Two proofs are
presented. The first proof applies a new combinatorial criterion for
Schur-positivity, while the second is bijective. The coefficients in the Schur
expansion are derived, and interpreted in terms of Bessel polynomials. We
present a Knuth-like equivalence relation on matchings, and show that every
equivalence class corresponds to an irreducible representation. We proceed to
find various refined Schur-positive sets, including the set of matchings with a
prescribed crossing number and the set of matchings with a given number of
pairs of intersecting chords. Finally, we characterize all the matchings $m$
such that the set of matchings avoiding $m$ is Schur-positive. |
| doi_str_mv | 10.48550/arxiv.2307.09894 |
| format | Article |
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is Schur-positive with respect to the set of short chords. Two proofs are
presented. The first proof applies a new combinatorial criterion for
Schur-positivity, while the second is bijective. The coefficients in the Schur
expansion are derived, and interpreted in terms of Bessel polynomials. We
present a Knuth-like equivalence relation on matchings, and show that every
equivalence class corresponds to an irreducible representation. We proceed to
find various refined Schur-positive sets, including the set of matchings with a
prescribed crossing number and the set of matchings with a given number of
pairs of intersecting chords. Finally, we characterize all the matchings $m$
such that the set of matchings avoiding $m$ is Schur-positive.</description><identifier>DOI: 10.48550/arxiv.2307.09894</identifier><language>eng</language><subject>Mathematics - Combinatorics</subject><creationdate>2023-07</creationdate><rights>http://arxiv.org/licenses/nonexclusive-distrib/1.0</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>228,230,776,881</link.rule.ids><linktorsrc>$$Uhttps://arxiv.org/abs/2307.09894$$EView_record_in_Cornell_University$$FView_record_in_$$GCornell_University$$Hfree_for_read</linktorsrc><backlink>$$Uhttps://doi.org/10.48550/arXiv.2307.09894$$DView paper in arXiv$$Hfree_for_read</backlink></links><search><creatorcontrib>Marmor, Avichai</creatorcontrib><title>Schur-Positivity of Short Chords in Matchings</title><description>We prove that the set of matchings with a fixed number of unmatched vertices
is Schur-positive with respect to the set of short chords. Two proofs are
presented. The first proof applies a new combinatorial criterion for
Schur-positivity, while the second is bijective. The coefficients in the Schur
expansion are derived, and interpreted in terms of Bessel polynomials. We
present a Knuth-like equivalence relation on matchings, and show that every
equivalence class corresponds to an irreducible representation. We proceed to
find various refined Schur-positive sets, including the set of matchings with a
prescribed crossing number and the set of matchings with a given number of
pairs of intersecting chords. Finally, we characterize all the matchings $m$
such that the set of matchings avoiding $m$ is Schur-positive.</description><subject>Mathematics - Combinatorics</subject><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2023</creationdate><recordtype>article</recordtype><sourceid>GOX</sourceid><recordid>eNotzr1uwjAUhmEvDBVwAZ3qG3B6HB_7OCOK-ieBqAR7ZBKbWIKkctII7r4tZfne7dPD2KOEDK3W8OzSJU5ZroAyKGyBD0zs6vY7ic9-iGOc4njlfeC7tk8jL3-3GXjs-MaNdRu747Bgs-BOg1_eO2f715d9-S7W27ePcrUWzhAKko1SKliJUtcAjoLTSHSQ3h5syImMBG-tNxo8aEQKSMbUDrHIJTZBzdnT_-3NW32leHbpWv25q5tb_QDwHTuI</recordid><startdate>20230719</startdate><enddate>20230719</enddate><creator>Marmor, Avichai</creator><scope>AKZ</scope><scope>GOX</scope></search><sort><creationdate>20230719</creationdate><title>Schur-Positivity of Short Chords in Matchings</title><author>Marmor, Avichai</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-a674-71d333f81415c00a7fa5477b1e8b8f277610e88e650e05447f4766ca449214df3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2023</creationdate><topic>Mathematics - Combinatorics</topic><toplevel>online_resources</toplevel><creatorcontrib>Marmor, Avichai</creatorcontrib><collection>arXiv Mathematics</collection><collection>arXiv.org</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Marmor, Avichai</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Schur-Positivity of Short Chords in Matchings</atitle><date>2023-07-19</date><risdate>2023</risdate><abstract>We prove that the set of matchings with a fixed number of unmatched vertices
is Schur-positive with respect to the set of short chords. Two proofs are
presented. The first proof applies a new combinatorial criterion for
Schur-positivity, while the second is bijective. The coefficients in the Schur
expansion are derived, and interpreted in terms of Bessel polynomials. We
present a Knuth-like equivalence relation on matchings, and show that every
equivalence class corresponds to an irreducible representation. We proceed to
find various refined Schur-positive sets, including the set of matchings with a
prescribed crossing number and the set of matchings with a given number of
pairs of intersecting chords. Finally, we characterize all the matchings $m$
such that the set of matchings avoiding $m$ is Schur-positive.</abstract><doi>10.48550/arxiv.2307.09894</doi><oa>free_for_read</oa></addata></record> |
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| subjects | Mathematics - Combinatorics |
| title | Schur-Positivity of Short Chords in Matchings |
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