The q -tangent and q -secant numbers via continued fractions
It is well known that the ( − 1 ) -evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct q -analogues of the latter result have been discovered by Foata and Han, and Josua...
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| Veröffentlicht in: | European journal of combinatorics 2010-10, Vol.31 (7), p.1689-1705 |
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| container_title | European journal of combinatorics |
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| creator | Shin, Heesung Zeng, Jiang |
| description | It is well known that the
(
−
1
)
-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct
q
-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some general continued fraction expansion formulae, which permit us to give a unified treatment of Josuat-Vergès’ two formulae and also to derive a new
q
-analogue of the aforementioned formulae. Our approach is based on a
(
p
,
q
)
-analogue of tangent and secant numbers via continued fractions and also the generating function of permutations with respect to the quintuple statistic consisting of fixed point number, weak excedance number, crossing number, nesting number and inversion number. We also give a combinatorial proof of Josuat-Vergès’ formulae by using a new linear model of derangements. |
| doi_str_mv | 10.1016/j.ejc.2010.04.003 |
| format | Article |
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(
−
1
)
-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct
q
-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some general continued fraction expansion formulae, which permit us to give a unified treatment of Josuat-Vergès’ two formulae and also to derive a new
q
-analogue of the aforementioned formulae. Our approach is based on a
(
p
,
q
)
-analogue of tangent and secant numbers via continued fractions and also the generating function of permutations with respect to the quintuple statistic consisting of fixed point number, weak excedance number, crossing number, nesting number and inversion number. We also give a combinatorial proof of Josuat-Vergès’ formulae by using a new linear model of derangements.</description><identifier>ISSN: 0195-6698</identifier><identifier>EISSN: 1095-9971</identifier><identifier>DOI: 10.1016/j.ejc.2010.04.003</identifier><language>eng</language><publisher>Elsevier Ltd</publisher><subject>Combinatorial analysis ; Combinatorics ; Inversions ; Mathematical models ; Mathematics ; Nesting ; Permutations ; Proving ; Statistics ; Tangents</subject><ispartof>European journal of combinatorics, 2010-10, Vol.31 (7), p.1689-1705</ispartof><rights>2010 Elsevier Ltd</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c364t-a76379fcd21380c1dc68f038e42bacb3870421b9387e033c9e67790f1ab64f853</citedby><cites>FETCH-LOGICAL-c364t-a76379fcd21380c1dc68f038e42bacb3870421b9387e033c9e67790f1ab64f853</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://www.sciencedirect.com/science/article/pii/S0195669810000491$$EHTML$$P50$$Gelsevier$$Hfree_for_read</linktohtml><link.rule.ids>230,314,776,780,881,3536,27903,27904,65309</link.rule.ids><backlink>$$Uhttps://hal.science/hal-00863434$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Shin, Heesung</creatorcontrib><creatorcontrib>Zeng, Jiang</creatorcontrib><title>The q -tangent and q -secant numbers via continued fractions</title><title>European journal of combinatorics</title><description>It is well known that the
(
−
1
)
-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct
q
-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some general continued fraction expansion formulae, which permit us to give a unified treatment of Josuat-Vergès’ two formulae and also to derive a new
q
-analogue of the aforementioned formulae. Our approach is based on a
(
p
,
q
)
-analogue of tangent and secant numbers via continued fractions and also the generating function of permutations with respect to the quintuple statistic consisting of fixed point number, weak excedance number, crossing number, nesting number and inversion number. We also give a combinatorial proof of Josuat-Vergès’ formulae by using a new linear model of derangements.</description><subject>Combinatorial analysis</subject><subject>Combinatorics</subject><subject>Inversions</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Nesting</subject><subject>Permutations</subject><subject>Proving</subject><subject>Statistics</subject><subject>Tangents</subject><issn>0195-6698</issn><issn>1095-9971</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2010</creationdate><recordtype>article</recordtype><recordid>eNp9UEFOwzAQtBBIlMIDuOUIh4R1nDq24FIhoEiVuJSz5Tgb6ih1WjupxO9xFcSR086sZnY1Q8gthYwC5Q9thq3JcogcigyAnZEZBblIpSzpOZkBjZhzKS7JVQgtAKULxmbkabPF5JCkg3Zf6IZEu_pEAxodmRt3FfqQHK1OTO8G60ask8ZrM9jehWty0egu4M3vnJPP15fN8ypdf7y9Py_XqWG8GFJdclbKxtQ5ZQIMrQ0XDTCBRV5pUzFRQpHTSkaAwJiRyMtSQkN1xYtGLNic3E93t7pTe2932n-rXlu1Wq7VaQcgOCtYcaRRezdp974_jBgGtbPBYNdph_0YFM0FKyGGz6OUTlLj-xA8Nn-3KahTq6pVsVV1alVBEb-w6HmcPBjzHi16FYxFZ7C2Hs2g6t7-4_4B3998mQ</recordid><startdate>20101001</startdate><enddate>20101001</enddate><creator>Shin, Heesung</creator><creator>Zeng, Jiang</creator><general>Elsevier Ltd</general><general>Elsevier</general><scope>6I.</scope><scope>AAFTH</scope><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><scope>1XC</scope></search><sort><creationdate>20101001</creationdate><title>The q -tangent and q -secant numbers via continued fractions</title><author>Shin, Heesung ; Zeng, Jiang</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c364t-a76379fcd21380c1dc68f038e42bacb3870421b9387e033c9e67790f1ab64f853</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2010</creationdate><topic>Combinatorial analysis</topic><topic>Combinatorics</topic><topic>Inversions</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Nesting</topic><topic>Permutations</topic><topic>Proving</topic><topic>Statistics</topic><topic>Tangents</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Shin, Heesung</creatorcontrib><creatorcontrib>Zeng, Jiang</creatorcontrib><collection>ScienceDirect Open Access Titles</collection><collection>Elsevier:ScienceDirect:Open Access</collection><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><collection>Hyper Article en Ligne (HAL)</collection><jtitle>European journal of combinatorics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Shin, Heesung</au><au>Zeng, Jiang</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The q -tangent and q -secant numbers via continued fractions</atitle><jtitle>European journal of combinatorics</jtitle><date>2010-10-01</date><risdate>2010</risdate><volume>31</volume><issue>7</issue><spage>1689</spage><epage>1705</epage><pages>1689-1705</pages><issn>0195-6698</issn><eissn>1095-9971</eissn><abstract>It is well known that the
(
−
1
)
-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). Recently, two distinct
q
-analogues of the latter result have been discovered by Foata and Han, and Josuat-Vergès, respectively. In this paper, we will prove some general continued fraction expansion formulae, which permit us to give a unified treatment of Josuat-Vergès’ two formulae and also to derive a new
q
-analogue of the aforementioned formulae. Our approach is based on a
(
p
,
q
)
-analogue of tangent and secant numbers via continued fractions and also the generating function of permutations with respect to the quintuple statistic consisting of fixed point number, weak excedance number, crossing number, nesting number and inversion number. We also give a combinatorial proof of Josuat-Vergès’ formulae by using a new linear model of derangements.</abstract><pub>Elsevier Ltd</pub><doi>10.1016/j.ejc.2010.04.003</doi><tpages>17</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0195-6698 |
| ispartof | European journal of combinatorics, 2010-10, Vol.31 (7), p.1689-1705 |
| issn | 0195-6698 1095-9971 |
| language | eng |
| recordid | cdi_hal_primary_oai_HAL_hal_00863434v1 |
| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Combinatorial analysis Combinatorics Inversions Mathematical models Mathematics Nesting Permutations Proving Statistics Tangents |
| title | The q -tangent and q -secant numbers via continued fractions |
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