Related papers: Combinatorics on permutation tableaux of type $A$ …
Within this research, two combinatorial bijections using Young diagrams were studied. The first is a special case of a bijective correspondence between two classes of combinatorial objects. Its proof, based on Young diagrams, establishes…
We discuss counting problems linked to finite versions of Cantor's diagonal of infinite tableaux. We extend previous results of [2] by refining an equivalence relation that reduces significantly the exhaustive generation. New enumerative…
We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non-oriented map…
We answer some questions related to multiplicity formulas by Rosenthal and Zelevinsky and by Lakshmibai and Weyman for points on Schubert varieties in Grassmannians. In particular, we give combinatorial interpretations in terms of…
In this work we introduce and study tree-like tableaux, which are certain fillings of Ferrers diagrams in simple bijection with permutation tableaux and alternative tableaux. We exhibit an elementary insertion procedure on our tableaux…
We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for…
We define mosaics, which are naturally in bijection with Knutson-Tao puzzles. We define an operation on mosaics, which shows they are also in bijection with Littlewood-Richardson skew-tableaux. Another consequence of this construction is…
This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called…
This paper establishes an analogue of the Robinson--Schensted correspondence for cylindric tableaux. In particular, for any pair of positive integers $(d,L)$, we construct a bijection between permutations that avoid the patterns $d\cdots 1…
We present bijections between four classes of combinatorial objects. Two of them, the class of unlabeled (2+2)-free posets and a certain class of involutions (or chord diagrams), already appeared in the literature, but were apparently not…
In this paper we present a combinatorial proof of a relation between the generating functions of unicellular and bicellular maps. This relation is a consequence of the Schwinger-Dyson equation of matrix theory. Alternatively it can be…
Linked partitions are introduced by Dykema in the study of transforms in free probability theory, whereas permutation tableaux are introduced by Steingr\'{i}msson and Williams in the study of totally positive Grassmannian cells. Let…
We study the factorizations of the permutation $(1,2,...,n)$ into $k$ factors of given cycle types. Using representation theory, Jackson obtained for each $k$ an elegant formula for counting these factorizations according to the number of…
Many important statistics of signed permutations are realized in the corresponding permutation tableaux or bare tableaux of type $B$: Alignments, crossings and inversions of signed permutations are realized in the corresponding permutation…
The Bitableax correspondence isomorphism/Koszul map Theorem (BCK Theorem, for short, Theorem 6.5 below) describes a relevant pair of mutually inverse vector space isomorphisms, the Koszul map K : U(gl(n))-> Sym(gl(n)) and the bitableaux…
We introduce the notion of "type" of a tableau, that allows us to define new families of tableaux including both balanced and standard Young tableaux. We use these new objects to describe the set of reduced decompositions of any…
We discuss permutation representations which are obtained by the natural action of $S_n \times S_n$ on some special sets of invertible matrices, defined by simple combinatorial attributes. We decompose these representations into…
We present a surprisingly new connection between two well-studied combinatorial classes: rooted connected chord diagrams on one hand, and rooted bridgeless combinatorial maps on the other hand. We describe a bijection between these two…
Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2].…
We develop several combinatorial models that are useful in the study of the $SL_n$-variety $\mathcal{X}$ of complete quadrics. Barred permutations parameterize the fixed points of the action of a maximal torus $T$ of $SL_n$, while…