Descent Statistic Research Articles

The descent statistic on involutions is not log-concave

We establish a combinatorial connection between the sequence ( i n , k ) counting the involutions on n letters with k descents and the sequence ( a n , k ) enumerating the semistandard Young tableaux on n cells with k symbols. This allows us to show that the sequences ( i n , k ) are not log-concave for some values of n , hence answering a conjecture due to F. Brenti.

Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams

The graph of zigzag diagrams is a close relative of Young's lattice. The boundary problem for this graph amounts to describing coherent random permutations with descent-set statistic, and is also related to certain positive characters on the algebra of quasi-symmetric functions. We establish connections to some further relatives of Young's lattice and solve the boundary problem by reducing it to the classification of spreadable total orders on integers, as recently obtained by Jacka and Warren.

A Probabilistic Approach to the Descent Statistic

We present a probabilistic approach to studying the descent statistic based upon a two-variable probability density. This density is log concave and, in fact, satisfies a higher order concavity condition. From these properties we derive quadratic inequalities for the descent statistic. Using Fourier series, we give exact expressions for the Euler numbers and the alternating r-signed permutations. We also obtain a probabilistic interpretation of the sin function.

Maximizing the descent statistic

For a subsetS, let the descent statistic β(S) be the number of permutations that have descent setS. We study inequalities between the descent statistics of subsets. Each subset (and its complement) is encoded by a list containing the lengths of the runs. We define two preorders that compare different lists based on the descent statistic. Using these preorders, we obtain a complete order on lists of the form (k i,P,k n−i, whereP is a palindrome, whose first entry is larger thank. We prove a conjecture due to Gessel, which determines the list that maximizes the descent statistic, among lists of a given size and given length. We also have a generalization of the boustrophedon transform of Millar, Sloane and Young.

Descents and one-dimensional characters for classical Weyl groups

This paper examine all sums of the form σ πϵW χ(π)t d(π) where W is a classical Weyl group, X is a one-dimensional character of W, and d( π) is the descent statistic. This completes a picture which is known when W is the symmetric group S n (the Weyl group A n−1 ). Surprisingly, the answers turn out to be simpler and generalize further for the other classical Weyl groups B n(≊C n) and D n . The B n, case uses sign-reversing involutions, while the D n case follows from a result of independent interest relating statistics for all three groups.

A decomposition of 2-weak vertex-packing polytopes

The 2-weak vertex-packing polytope of a loopless graphG withd vertices is the subset of the unitd-cube satisfyingxi+xj≤1 for every edge (i,j) ofG. The dilation by 2 of this polytope is a polytope[Figure not available: see fulltext.] with integral vertices. We triangulate[Figure not available: see fulltext.] with lattice simplices of minimal volume and label the maximal simplices with elements of the hyperoctahedral groupBd. This labeling gives rise to a shelling of the triangulation[Figure not available: see fulltext.] of[Figure not available: see fulltext.], where theh-vector of[Figure not available: see fulltext.] (and the Ehrharth*-vector of[Figure not available: see fulltext.] can be computed as a descent statistic on a subset ofBd defined in terms ofG. A recursive way of computing theh-vector of[Figure not available: see fulltext.] is also given, and a recursive formula for the volume of[Figure not available: see fulltext.].