TY - JOUR

T1 - Signed countings of types B and D permutations and t,q-Euler numbers

AU - Eu, Sen Peng

AU - Fu, Tung Shan

AU - Hsu, Hsiang Chun

AU - Liao, Hsin Chieh

N1 - Funding Information:
The authors thank the referees for reading the manuscript carefully and providing helpful suggestions. This research is partially supported by Ministry of Science and Technology (MOST) , Taiwan, under grants 104-2115-M-003-014-MY3 (S.-P. Eu), 105-2115-M-153-002-MY2 (T.-S. Fu) and MOST postdoctoral fellowship 106-2811-M003-011 (H.-C. Hsu).

PY - 2018/6

Y1 - 2018/6

N2 - It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length n is the Euler number En, alternating in sign, if n is odd (even, respectively). Josuat-Vergès obtained a q-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès and Kim. Springer numbers are analogous Euler numbers that count the alternating permutations of type B, called snakes. Josuat-Vergès derived bivariate polynomials Qn(t,q) and Rn(t,q) as generalized Euler numbers via successive q-derivatives and multiplications by t on polynomials in t. The other goal in this paper is to give a combinatorial interpretation of Qn(t,q) and Rn(t,q) as the enumerators of the snakes with restrictions.

AB - It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length n is the Euler number En, alternating in sign, if n is odd (even, respectively). Josuat-Vergès obtained a q-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès and Kim. Springer numbers are analogous Euler numbers that count the alternating permutations of type B, called snakes. Josuat-Vergès derived bivariate polynomials Qn(t,q) and Rn(t,q) as generalized Euler numbers via successive q-derivatives and multiplications by t on polynomials in t. The other goal in this paper is to give a combinatorial interpretation of Qn(t,q) and Rn(t,q) as the enumerators of the snakes with restrictions.

KW - Continued fractions

KW - Derangements

KW - Euler number

KW - Signed permutations

KW - Springer number

KW - Weighted bicolored Motzkin paths

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U2 - 10.1016/j.aam.2018.02.004

DO - 10.1016/j.aam.2018.02.004

M3 - Article

AN - SCOPUS:85042685052

VL - 97

SP - 1

EP - 26

JO - Advances in Applied Mathematics

JF - Advances in Applied Mathematics

SN - 0196-8858

ER -