20 research outputs found
Cyclic Eulerian Elements
AbstractLetSnbe the symmetric group on {1,…,n} and Q[Sn] its group algebra over the rational field; we assumen≥2. π∈Snis said a descent ini, 1≤i≤n-1, if π(i)>π (i+1); moreover, π is said to have a cyclic descent if π(n)>π(1). We define the cyclic Eulerian elements as the sums of all elements inSnhaving a fixed global number of descents, possibly including the cyclic one. We show that the cyclic Eulerian elements linearly span a commutative semisimple algebra of Q[Sn], which is naturally isomorphic to the algebra of the classical Eulerian elements. Moreover, we give a complete set of orthogonal idempotents for such algebra, which are strictly related to the usual Eulerian idempotents
A Cyclic Analogue of Stanley's Shuffle Theorem
We introduce the cyclic major index of a cycle permutation and give a
bivariate analogue of enumerative formula for the cyclic shuffles with a given
cyclic descent numbers due to Adin, Gessel, Reiner and Roichman, which can be
viewed as a cyclic analogue of Stanley's Shuffle Theorem. This gives an answer
to a question of Adin, Gessel, Reiner and Roichman, which has been posed by
Domagalski, Liang, Minnich, Sagan, Schmidt and Sietsema again.Comment: 7 pages. We thank Bruce Sagan for providing useful comments and
relevant references for the earlier versio
Affine shuffles, shuffles with cuts, the Whitehouse module, and patience sorting
Type A affine shuffles are compared with riffle shuffles followed by a cut.
Although these probability measures on the symmetric group S_n are different,
they both satisfy a convolution property. Strong evidence is given that when
the underlying parameter satisfies , the induced measures on
conjugacy classes of the symmetric group coincide. This gives rise to
interesting combinatorics concerning the modular equidistribution by major
index of permutations in a given conjugacy class and with a given number of
cyclic descents. It is proved that the use of cuts does not speed up the
convergence rate of riffle shuffles to randomness. Generating functions for the
first pile size in patience sorting from decks with repeated values are
derived. This relates to random matrices.Comment: Galley version for J. Alg.; minor revisions in Sec.
Affine descents and the Steinberg torus
Let be an irreducible affine Weyl group with Coxeter complex
, where denotes the associated finite Weyl group and the
translation subgroup. The Steinberg torus is the Boolean cell complex obtained
by taking the quotient of by the lattice . We show that the
ordinary and flag -polynomials of the Steinberg torus (with the empty face
deleted) are generating functions over for a descent-like statistic first
studied by Cellini. We also show that the ordinary -polynomial has a
nonnegative -vector, and hence, symmetric and unimodal coefficients. In
the classical cases, we also provide expansions, identities, and generating
functions for the -polynomials of Steinberg tori.Comment: 24 pages, 2 figure
Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups
We use the method of mutual interlacing to prove two conjectures on the
real-rootedness of Eulerian-like polynomials: Brenti's conjecture on
-Eulerian polynomials for Weyl groups of type , and Dilks, Petersen, and
Stembridge's conjecture on affine Eulerian polynomials for irreducible finite
Weyl groups.
For the former, we obtain a refinement of Brenti's -Eulerian polynomials
of type , and then show that these refined Eulerian polynomials satisfy
certain recurrence relation. By using the Routh--Hurwitz theory and the
recurrence relation, we prove that these polynomials form a mutually
interlacing sequence for any positive , and hence prove Brenti's conjecture.
For , our result reduces to the real-rootedness of the Eulerian
polynomials of type , which were originally conjectured by Brenti and
recently proved by Savage and Visontai.
For the latter, we introduce a family of polynomials based on Savage and
Visontai's refinement of Eulerian polynomials of type . We show that these
new polynomials satisfy the same recurrence relation as Savage and Visontai's
refined Eulerian polynomials. As a result, we get the real-rootedness of the
affine Eulerian polynomials of type . Combining the previous results for
other types, we completely prove Dilks, Petersen, and Stembridge's conjecture,
which states that, for every irreducible finite Weyl group, the affine descent
polynomial has only real zeros.Comment: 28 page