219 research outputs found
Riordan Paths and Derangements
Riordan paths are Motzkin paths without horizontal steps on the x-axis. We
establish a correspondence between Riordan paths and
-avoiding derangements. We also present a combinatorial proof
of a recurrence relation for the Riordan numbers in the spirit of the
Foata-Zeilberger proof of a recurrence relation on the Schr\"oder numbers.Comment: 9 pages, 2 figure
Minimal factorizations of permutations into star transpositions
We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form (excluded due to format error) source. This generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored. Our result exhibits an unexpected and simple symmetry of star factorizations that has yet to be explained in a satisfactory manner
Stable multivariate -Eulerian polynomials
We prove a multivariate strengthening of Brenti's result that every root of
the Eulerian polynomial of type is real. Our proof combines a refinement of
the descent statistic for signed permutations with the notion of real
stability-a generalization of real-rootedness to polynomials in multiple
variables. The key is that our refined multivariate Eulerian polynomials
satisfy a recurrence given by a stability-preserving linear operator. Our
results extend naturally to colored permutations, and we also give stable
generalizations of recent real-rootedness results due to Dilks, Petersen, and
Stembridge on affine Eulerian polynomials of types and . Finally,
although we are not able to settle Brenti's real-rootedness conjecture for
Eulerian polynomials of type , nor prove a companion conjecture of Dilks,
Petersen, and Stembridge for affine Eulerian polynomials of types and ,
we indicate some methods of attack and pose some related open problems.Comment: 17 pages. To appear in J. Combin. Theory Ser.
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