無中文摘要

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.

[1] V.I. Arnol'd, The calculus of snakes and the combinatorics of Bernoulli, Euler, and Springer numbers for Coxeter groups, Russian Math. Surveys 47 (1992) 1--51.

[2] C.A. Athanasiadis, Edgewise subdivisions, local $h$-polynomials and excedances in the wreath product $\mathbb{Z}_r \wr \mathfrak{S}_n$, SIAM J. Discrete Math. 28 (2014) 1479--1492.

[3] D. Chebikin, Variations on descents and inversions in permutations, Electron. J. Combin. 15 (2008) \#R132.

[4] S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math. 38(2) (2007) 149--163.

[5] S. Corteel, M. Josuat-Verg\`es, J.S. Kim, Crossings of signed permutations and $q$-Eulerian
numbers of type B, J. Combin. 4(2) (2013) 191--228.

[6] S. Corteel, M. Josuat-Verg\`es, L.K. Williams, Matrix Ansatz, orthogonal polynomials and
permutations, Adv. Appl. Math. 46 (2011) 209--225.

[7] S. Cho, K. Park, A Combinatorial Proof of a Symmetry of $(t,q)$-Eulerian Numbers of Type B and Type D, Ann. Comb. 22 (2018) 99--134.

[8] S.-P. Eu, T.-S. Fu, H. -C. Hsu, H.-C. Liao, Signed countings of types B and D permutations and $t,q$-Euler numbers, Adv. Appl. Math. 97 (2018) 1-26.

[9] L. Euler, Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, in \textit{Academiae Imperialis Scientiarum Petropolitanae}, St. Petersburg 1755 (Part II, chapter 7: Methodus summandi superior ulterius promota.)

[10] P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math. 32 (1980) 125--161.

[11] D. Foata, G.-N. Han, The $q$-tangent and $q$-secant numbers via basic eulerian polynomials, Proc. Amer. Math. Soc. 138(2)(2010) 385--393.

[12] D. Foata, M.-P. Sch\"utzenberger,
Th\'eorie g\'eom\'etrique des polyn\^omes eul\'eriens, in: Lecture Notes in Mathematics vol. 138, Springer-Verlag, Berlin, 1970.

[13] J. Fran\c con, X. Viennot, Permutations selon leurs pics, creux, doubles mont\'ees et double descentes, nombres d'Euler et nombres de Genocchi, Discrete Math. 28 (1979) 21--35.

[14] D. Foata, D. Zeilberger, Denert's permutation statistic is indeed Euler-Mahonian, Stud. Appl. Math. 83(1) (1990) 31--59.

[15] A. Hamdi, Symmetric distribution of crossings and nestings in permutations of type B, Electron. J. Combin. 18 (2011), P200.

[16] G.-N. Han, A. Randrianarivony, J. Zeng, Un autre q-analogue des nombres d'Euler, S\'{e}m. Lothar. Combin, B42e (1999).

[17] M.E. Hoffman, Derivative polynomials, Euler polynomials, and associated integer sequences,
Electron. J. Combin. 6 (1999), R21.

[18] M. Josuat-Verg\`{e}s, A $q$-enumeration of alternating permutations, European J. Combin. 31 (2010) 1892--1906.

[19] M. Josuat-Verg\`es, Enumeration of snakes and cycle-alternating permutations, Australas. J. Combin. 60(3) (2014) 279--305.

[20] P.D. Roselle, Permutations by number of rises and successions, Proc. Amer. Math. Soc. 19 (1968) 8--16.

[21] H. Shin, J. Zeng, The $q$-tangent and $q$-secant numbers via continued fractions, European J. Combin. 31 (2010) 1689--1705.

[22] H. Shin, J. Zeng, Symmetric unimodal expansions of excedances in colored permutations, European J. Combin. 52, part A (2016) 174--196.

[23] T.A. Springer, Remarks on a combinatorial problem, Nieuw Arch. Wisk. 19(3) (1971), 30–36.

[24] R. P. Stanley, Enumerative Combinatorics. Vol. 1, vol. 49 of \textit{Cambridge Studies in Adcanced Mathematics.} Cambridge University Press, Cambridge, 1997.

[25] L.K. Williams, Enumeration of totally positive Grassmann cells, Adv. Math. 190 (2005) 319--342.