Annals of Combinatorics

Chu, Wenchang; di Claudio, Leontina

doi: 10.1007/s00026-005-0264-4pmid: N/A

By means of matrix decomposition and the partial fraction method, we establish several determinant evaluation formulas, which can be considered as generalizations of the Vandermonde and Cauchy determinants.

Hetyei, Gábor

doi: 10.1007/s00026-005-0265-3pmid: N/A

We construct spherical CW-complexes whose face structure may be conveniently described using a system of polyspherical coordinates introduced by Vilenkin, Kuznetsov and Smorodinskii. We show that these complexes may be constructed by repeated use of CW-suspension, free join, and edge subdivision. We prove that all CW-spheres constructed in this way have non-negative cd-index and thus verify Stanley’s famous conjecture. Among the particular examples we find a new class of partially ordered sets whose order complexes encode the derivative polynomials for secant of even degree. The geometric constructions presented in this paper generalize CW-complexes introduced whose flag numbers are suitable to encode systems of orthogonal polynomials.

Krob, D.; Thibon, J.

doi: 10.1007/s00026-005-0266-2pmid: N/A

Using the theory of noncommutative symmetric functions, we introduce the higher order peak algebras (Sym(N)) N≥1, a sequence of graded Hopf algebras which contain the descent algebra and the usual peak algebra as initial cases (N=1 and N=2). We compute their Hilbert series, introduce and study several combinatorial bases, and establish various algebraic identities related to the multisection of formal power series with noncommutative coefficients.

Martini, Horst; Wenzel, Walter

doi: 10.1007/s00026-005-0267-1pmid: N/A

For an arbitrary set E and a given closure operator $$ \sigma :\,\mathcal{P}(E) \to \,\mathcal{P}(E) $$ , we want to construct a symmetric closure operator $$ \ifmmode\expandafter\hat\else\expandafter\^\fi{\upsigma }:\,\mathcal{P}(E) \to \,\mathcal{P}(E) $$ via some – possibly infinite – iteration process. If E is finite, the corresponding symmetric closure operator . $$ {\ifmmode\expandafter\hat\else\expandafter\^\fi{\upsigma }} $$ defines a matroid. If $$ E = \,\mathbb{R}^{n} $$ and $$ \upsigma$$ is the convex closure operator, $$ {\ifmmode\expandafter\hat\else\expandafter\^\fi{\upsigma }} $$ turns out to be the affine closure operator. Moreover, we apply the symmetrization process to closure operators induced by visibility.

Rhoades, Brendon; Skandera, Mark

doi: 10.1007/s00026-005-0268-0pmid: N/A

We use the Temperley-Lieb algebra to define a family of totally nonnegative polynomials of the form $$ \Sigma _{{\upsigma \in S_{n} }} f(\upsigma )x_{{1,\upsigma (1)}} \cdots x_{{n,\upsigma (n)}} $$ . The cone generated by these polynomials contains all totally nonnegative polynomials of the form $$ \Delta _{{J,J' }} (x)\Delta _{{L,L' }} (x) - \Delta _{{I,I' }} (x)\Delta _{{K,K' }} (x) $$ , where, $$ \Delta _{{I,I' }} (x), \ldots ,\Delta _{{K,K' }} (x) $$ are matrix minors. We also give new conditions on the sets I,...,K′ which characterize differences of products of minors which are totally nonnegative.

Timofte, Vlad

doi: 10.1007/s00026-005-0269-zpmid: N/A

We give a criterion for bases of the ring of symmetric functions in n indeterminates over a commutative ring R with identity. A related algorithm is presented in the last section.