4,051 research outputs found
Positivity results for Stanley's character polynomials
Stanley introduced expressions for the normalized characters of the symmetric
group and stated some positivity conjectures for these expressions. Here, we
give an affirmative partial answer to Stanley's positivity conjectures about
the expressions using results on Kerov polynomials. In particular, we use new
positivity results by Goulden and the present author. We shall see that the
generating series introduced by them is critical to our discussion.Comment: 20 pages, 2 figures, v2, minor revisions, fixed typos et
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
Stanley's character polynomials and coloured factorisations in the symmetric group
In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, SĂŠm. Lothar. Combin. 50 (2003) B50d, 11 p.] the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [R.P. Stanley, A conjectured combinatorial interpretation of the normalised irreducible character values of the symmetric group, math.CO/0606467, 2006] the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree
The number of lattice paths below a cyclically shifting boundary
We count the number of lattice paths lying under a cyclically shifting piecewise linear boundary of varying slope. Our main result can be viewed as an extension of well-known enumerative formulae concerning lattice paths dominated by lines of integer slope (e.g. the generalized ballot theorem). Its proof is bijective, involving a classical âreflectionâ argument. Moreover, a straightforward refinement of our bijection allows for the counting of paths with a specified number of corners. We also show how the result can be applied to give elegant derivations for the number of lattice walks under certain periodic boundaries. In particular, we recover known expressions concerning paths dominated by a line of half-integer slope, and some new and old formulae for paths lying under special âstaircases.
An explicit form for Kerov's character polynomials
Kerov considered the normalized characters of irreducible representations of
the symmetric group, evaluated on a cycle, as a polynomial in free cumulants.
Biane has proved that this polynomial has integer coefficients, and made
various conjectures. Recently, Sniady has proved Biane's conjectured explicit
form for the first family of nontrivial terms in this polynomial. In this
paper, we give an explicit expression for all terms in Kerov's character
polynomials. Our method is through Lagrange inversion.Comment: 17 pages, 1 figur
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