546 research outputs found
Specializations and Extensions of the quantum MacMahon Master Theorem
We study some specializations and extensions of the quantum version of the
MacMahon Master Theorem derived by Garoufalidis, Le and Zeilberger. In
particular, we obtain a (t,q)-analogue for the Cartier-Foata noncommutative
version and a semi-strong (t,q)-analogue for the contextual algebra.Comment: 12 page
The symmetric and unimodal expansion of Eulerian polynomials via continued fractions
This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J.
Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the
coefficients in an expansion of generalized Eulerian polynomials, which implies
the symmetric and unimodal property of the Eulerian numbers. We show that such
a formula with the conjectured property can be derived from the combinatorial
theory of continued fractions. We also discuss an analogous expansion for the
corresponding formula for derangements and prove a -analogue of the fact
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). The -analogue unifies and generalizes our recent
results (European J. Combin. \textbf{31} (2010), no.~7, 1689--1705.) and that
of Josuat-Verg\`es (European J. Combin. \textbf{31} (2010), no.~7, 1892--1906).Comment: 19 pages, 2 figure
Fix-Mahonian Calculus, I: two transformations
We construct two bijections of the symmetric group S_n onto itself that
enable us to show that three new three-variable statistics are equidistributed
with classical statistics involving the number of fixed points. The first one
is equidistributed with the triplet (fix,des,maj), the last two with
(fix,exc,maj), where "fix," "des," "exc" and "maj" denote the number of fixed
points, the number of descents, the number of excedances and the major index,
respectively
Fix-Mahonian Calculus, II: further statistics
Using classical transformations on the symmetric group and two
transformations constructed in Fix-Mahonian Calculus I, we show that several
multivariable statistics are equidistributed either with the triplet
(fix,des,maj), or the pair (fix,maj), where "fix," "des" and "maj" denote the
number of fixed points, the number of descents and the major index,
respectively
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