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

 dc.contributor 游森棚 zh_TW dc.contributor Eu, Sen-Peng en_US dc.contributor.author 廖信傑 zh_TW dc.contributor.author Liao, Hsin-Chieh en_US dc.date.accessioned 2019-09-05T01:06:32Z dc.date.available 2018-06-27 dc.date.available 2019-09-05T01:06:32Z dc.date.issued 2018 dc.description.abstract 無中文摘要 zh_TW dc.description.abstract A 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.identifier G060440034S dc.identifier.uri http://etds.lib.ntnu.edu.tw/cgi-bin/gs32/gsweb.cgi?o=dstdcdr&s=id=%22G060440034S%22.&%22.id.& dc.identifier.uri http://rportal.lib.ntnu.edu.tw:80/handle/20.500.12235/101575 dc.language 英文 dc.subject Signed permutations zh_TW dc.subject Euler numbers zh_TW dc.subject Springer numbers zh_TW dc.subject q-analogue zh_TW dc.subject continued fractions zh_TW dc.subject weighted bicolored Motzkin paths zh_TW dc.subject Signed permutations en_US dc.subject Euler numbers en_US dc.subject Springer numbers en_US dc.subject q-analogue en_US dc.subject continued fractions en_US dc.subject weighted bicolored Motzkin paths en_US dc.title Signed Countings of Type B and D Permutations and t,q-Euler numbers zh_TW dc.title Signed Countings of Type B and D Permutations and t,q-Euler numbers en_US

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