Signed Countings of Type B and D Permutations and t,q-Euler numbers

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dc.contributor游森棚zh_TW
dc.contributorEu, Sen-Pengen_US
dc.contributor.author廖信傑zh_TW
dc.contributor.authorLiao, Hsin-Chiehen_US
dc.date.accessioned2019-09-05T01:06:32Z
dc.date.available2018-06-27
dc.date.available2019-09-05T01:06:32Z
dc.date.issued2018
dc.description.abstract無中文摘要zh_TW
dc.description.abstractA classical result states that the parity balance of number of excedances of all permutations (derangements, respectivly) of length $n$ is the Euler number. In 2010, Josuat-Verg\`{e}s gives a $q$-analogue with $q$ representing the number of crossings. We extend this result to the permutations (derangements, respectively) of type B and D. It turns out that the signed counting are related to the derivative polynomials of $\tan$ and $\sec$. Springer numbers defined by Springer can be regarded as an analogue of Euler numbers defined on every Coxeter group. In 1992 Arnol'd showed that the Springer numbers of classical types A, B, D count various combinatorial objects, called snakes. In 1999 Hoffman found that derivative polynomials of $\sec x$ and $\tan x$ and their subtraction evaluated at certain values count exactly the number of snakes of certain types. Then Josuat-Verg\`{e}s studied the $(t,q)$-analogs of derivative polynomials $Q_n(t,q)$, $R_n(t,q)$ and showed that as setting $q=1$ the polynomials are enumerators of snakes with respect to the number of sign changing. Our second result is to find a combinatorial interpretations of $Q_n(t,q)$ and $R_n(t,q)$ as enumerator of the snakes, although the outcome is somewhat messy.en_US
dc.description.sponsorship數學系zh_TW
dc.identifierG060440034S
dc.identifier.urihttp://etds.lib.ntnu.edu.tw/cgi-bin/gs32/gsweb.cgi?o=dstdcdr&s=id=%22G060440034S%22.&%22.id.&
dc.identifier.urihttp://rportal.lib.ntnu.edu.tw:80/handle/20.500.12235/101575
dc.language英文
dc.subjectSigned permutationszh_TW
dc.subjectEuler numberszh_TW
dc.subjectSpringer numberszh_TW
dc.subjectq-analoguezh_TW
dc.subjectcontinued fractionszh_TW
dc.subjectweighted bicolored Motzkin pathszh_TW
dc.subjectSigned permutationsen_US
dc.subjectEuler numbersen_US
dc.subjectSpringer numbersen_US
dc.subjectq-analogueen_US
dc.subjectcontinued fractionsen_US
dc.subjectweighted bicolored Motzkin pathsen_US
dc.titleSigned Countings of Type B and D Permutations and t,q-Euler numberszh_TW
dc.titleSigned Countings of Type B and D Permutations and t,q-Euler numbersen_US

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