Related papers: Equality of P-partition generating functions
In this note we investigate mixed partitions with extra condition on the sizes of the blocks. We give a general formula and the generating function. We consider in more details a special case, determining the generating functions, some…
Elementary proofs are given for sums of Schur functions over partitions into at most n parts each less than or equal to m for which i) all parts are even, ii) all parts of the conjugate partition are even. Also, an elementary proof of a…
We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special…
We study two constructions related to the intervals of finite posets. The first one is a poset. The second one is more complicated. Loosely speaking it can be seen as a poset with some extra zero-relations. As main result, we show that…
We show the equivalence of the Pieri formula for flag manifolds and certain identities among the structure constants, giving new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function…
We wonder if there is a way to make all Schur functions in all representations equal. This is impossible for fixed value of time variables, but can be achieved for averages. It appears that the corresponding measure is just Gaussian in…
The Brenke type generating functions are the polynomial generating functions of the form $$\sum_{n=0}^{\infty}{P_n(x )\over n!}t^n=A(t)B(xt), $$ where $A$ and $B$ are two formal power series subject to the conditions…
The generating function for the number of purely crossing partitions of {1,...,n} is found in terms of the generating function for Bell numbers. Further results about generating functions for asymptotic moments of certain random Vandermonde…
We study generating functions in the context of Rota-Baxter algebras. We show that exponential generating functions can be naturally viewed in a very special case of complete free commutative Rota-Baxter algebras. This allows us to use free…
We define a poset of partitions associated to an operad. We prove that the operad is Koszul if and only if the poset is Cohen-Macaulay. In one hand, this characterisation allows us to compute the homology of the poset. This homology is…
Stanley's theory of $(P,\omega)$-partitions is a standard tool in combinatorics. It can be extended to allow for the presence of a restriction, that is a given maximal value for partitions at each vertex of the poset, as was shown by Assaf…
We introduce iposets---posets with interfaces---equipped with a novel gluing composition along interfaces and the standard parallel composition. We study their basic algebraic properties as well as the hierarchy of gluing-parallel posets…
We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation,…
Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for $k$-colored partition functions $p_{-k}(n)$ for all $k\geq2$. This enables us to extend the $k$-colored partition function multiplicatively to a…
The counting of partitions according to their genus is revisited. The case of genus 0 -- non-crossing partitions -- is well known. Our approach relies on two pillars: first a functional equation between generating functions, originally…
We prove analytic and combinatorial identities reminiscent of Schur's classical partition theorem. Specifically, we show that certain families of overpartitions whose parts satisfy gap conditions are equinumerous with partitions whose parts…
We study the generating function of the excess number of Rogers-Ramanujan partitions with odd rank over those with even rank, and, using combinatorial and analytical techniques, show that this generating function is closely connected with…
We introduce a family of quasisymmetric functions called {\em Eulerian quasisymmetric functions}, which have the property of specializing to enumerators for the joint distribution of the permutation statistics, major index and excedance…
Let $\nabla^\lambda$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^\lambda…
Using the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that…