AbstractA uniform approach to computing with infinite objects like real numbers, tuples of these, compacts sets and uniformly continuous maps is presented. In the work of Berger, it was shown how to extract certified algorithms working with the signed digit representation from constructive proofs. Berger and the present author generalised this approach to complete metric spaces and showed how to deal with compact sets. Here, we unify this work and lay the foundations for doing a similar thing for the much more comprehensive class of compact Hausdorff spaces occurring in applications. The approach is of the same computational power as Weihrauch’s Type-Two Theory of Effectivity.
We describe a combinatorial formula for the coefficients when the dual immaculate quasisymmetric func- tions are decomposed into Young quasisymmetric Schur functions. We prove this using an analogue of Schensted insertion. We also provide a Remmel-Whitney style rule to generate these coefficients algorithmically.
Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups $(GL_n)$. The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for Littlewood-Richardson coefficients corresponding to indices sufficiently far from the boundary of the Littlewood Richardson cone. 2. A proof of approximate log-concavity of the above mentioned class of Littlewood-Richardson coefficients.
The Classical Shuffle Conjecture of Haglund et al. (2005) has a symmetric function side and a combinatorial side. The combinatorial side $q,t$-enumerates parking functions in the $n ×n$ lattice. The symmetric function side may be simply expressed as $∇ e_n$ , where $∇$ is the Macdonald eigen-operator introduced by Bergeron and Garsia (1999) and $e_n$ is the elementary symmetric function. The combinatorial side has been extended to parking functions in the $m ×n$ lattice for coprime $m,n$ by Hikita (2012). Recently, Gorsky and Negut have been able to extend the Shuffle Conjecture by combining their work (2012a, 2012b, 2013) (related to work of Schiffmann and Vasserot (2011, 2013)) with Hikita's combinatorial results. We prove this new conjecture for the cases $m=2$ and $n=2$ .
In our previous works "Pfaffian decomposition and a Pfaffian analogue of $q$-Catalan Hankel determinants'' (by M.Ishikawa, H. Tagawa and J. Zeng, J. Combin. Theory Ser. A, 120, 2013, 1263-1284) we have proposed several ways to evaluate certain Catalan-Hankel Pffafians and also formulated several conjectures. In this work we propose a new approach to compute these Catalan-Hankel Pffafians using Selberg's integral as well as their $q$-analogues. In particular, this approach permits us to settle most of the conjectures in our previous paper.
We define the bigraphical arrangement of a graph and show that the Pak-Stanley labels of its regions are the parking functions of a closely related graph, thus proving conjectures of Duval, Klivans, and Martin and of Hopkins and Perkinson. A consequence is a new proof of a bijection between labeled graphs and regions of the Shi arrangement first given by Stanley. We also give bounds on the number of regions of a bigraphical arrangement. The full version of this paper is forthcoming in the $\textit{Transactions of the American Mathematical Society}$
We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $\alpha$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $\alpha$. Secondly, we describe asymptotically the shape of random Young diagrams under some deformation of Plancherel measure.
We generalize the brick polytope of V. Pilaud and F. Santos to spherical subword complexes for finite Coxeter groups. This construction provides polytopal realizations for a certain class of subword complexes containing all cluster complexes of finite types. For the latter, the brick polytopes turn out to coincide with the known realizations of generalized associahedra, thus opening new perspectives on these constructions. This new approach yields in particular the vertex description and a relevant Minkowski sum decomposition of generalized associahedra.
In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials.
A special case of Haiman's identity [Invent. Math. 149 (2002), pp. 371–407] for the character of the quotient ring of diagonal coinvariants under the diagonal action of the symmetric group yields a formula for the bigraded Hilbert series as a sum of rational functions in $q,t$. In this paper we show how a summation identity of Garsia and Zabrocki for Macdonald polynomial Pieri coefficients can be used to transform Haiman's formula for the Hilbert series into an explicit polynomial in $q,t$ with integer coefficients. We also provide an equivalent formula for the Hilbert series as the constant term in a multivariate Laurent series.
By investigating the action of the 0-Hecke algebra on the coinvariant algebra and the complete flag variety, we interpret generating functions counting the permutations with fixed inverse descent set by their inversion number and major index.
We present $\textit{type preserving}$ bijections between noncrossing and nonnesting partitions for all classical reflection groups, answering a question of Athanasiadis and Reiner. The bijections for the abstract Coxeter types $B$, $C$ and $D$ are new in the literature. To find them we define, for every type, sets of statistics that are in bijection with noncrossing and nonnesting partitions, and this correspondence is established by means of elementary methods in all cases. The statistics can be then seen to be counted by the generalized Catalan numbers Cat$(W)$ when $W$ is a classical reflection group. In particular, the statistics of type $A$ appear as a new explicit example of objects that are counted by the classical Catalan numbers.
The multiplihedra $\mathcal{M}_{\bullet} = (\mathcal{M}_n)_{n \geq 1}$ form a family of polytopes originating in the study of higher categories and homotopy theory. While the multiplihedra may be unfamiliar to the algebraic combinatorics community, it is nestled between two families of polytopes that certainly are not: the permutahedra $\mathfrak{S}_{\bullet}$ and associahedra $\mathcal{Y}_{\bullet}$. The maps $\mathfrak{S}_{\bullet} \twoheadrightarrow \mathcal{M}_{\bullet} \twoheadrightarrow \mathcal{Y}_{\bullet}$ reveal several new Hopf structures on tree-like objects nestled between the Hopf algebras $\mathfrak{S}Sym$ and $\mathcal{Y}Sym$. We begin their study here, showing that $\mathcal{M}Sym$ is a module over $\mathfrak{S}Sym$ and a Hopf module over $\mathcal{Y}Sym$. An elegant description of the coinvariants for $\mathcal{M}Sym$ over $\mathcal{Y}Sym$ is uncovered via a change of basis-using Möbius inversion in posets built on the $1$-skeleta of $\mathcal{M}_{\bullet}$. Our analysis uses the notion of an $\textit{interval retract}$ that should be of independent interest in poset combinatorics. It also reveals new families of polytopes, and even a new factorization of a known projection from the associahedra to hypercubes.
In this article we study a class of monoids that includes Garside monoids, and give a simple combinatorial proof of a formula for the formal sum of all elements of the monoid. This leads to a formula for the growth function of the monoid in the homogeneous case, and can also be lifted to a resolution of the monoid algebra. These results are then applied to known monoids related to Coxeter systems: we give the growth function of the Artin-Tits monoids, and do the same for the dual braid monoids. In this last case we show that the monoid algebras of the dual braid monoids of type A and B are Koszul algebras.
Sergey Kitaev, Jeffrey Liese, Jeffrey Remmel
et al.
Let $P$ be a partially ordered set and consider the free monoid $P^{\ast}$ of all words over $P$. If $w,w' \in P^{\ast}$ then $w'$ is a factor of $w$ if there are words $u,v$ with $w=uw'v$. Define generalized factor order on $P^{\ast}$ by letting $u \leq w$ if there is a factor $w'$ of $w$ having the same length as $u$ such that $u \leq w'$, where the comparison of $u$ and $w'$ is done componentwise using the partial order in $P$. One obtains ordinary factor order by insisting that $u=w'$ or, equivalently, by taking $P$ to be an antichain. Given $u \in P^{\ast}$, we prove that the language $\mathcal{F}(u)=\{w : w \geq u\}$ is accepted by a finite state automaton. If $P$ is finite then it follows that the generating function $F(u)=\sum_{w \geq u} w$ is rational. This is an analogue of a theorem of Björner and Sagan for generalized subword order. We also consider $P=\mathbb{P}$, the positive integers with the usual total order, so that $\mathbb{P}^{\ast}$ is the set of compositions. In this case one obtains a weight generating function $F(u;t,x)$ by substituting $tx^n$ each time $n \in \mathbb{P}$ appears in $F(u)$. We show that this generating function is also rational by using the transfer-matrix method. Words $u,v$ are said to be Wilf equivalent if $F(u;t,x)=F(v;t,x)$ and we can prove various Wilf equivalences combinatorially. Björner found a recursive formula for the Möbius function of ordinary factor order on $P^{\ast}$. It follows that one always has $\mu (u,w)=0, \pm 1$. Using the Pumping Lemma we show that the generating function $M(u)= \sum_{w \geq u} | \mu (u,w) | w$ can be irrational.