Related papers: Combinatorics for Certain Skew Young Tableaux, Dyc…
In this series of articles we study connections between combinatorics of multidimensional generalizations of Cauchy identity and continuous objects such as multidimensional Brownian motions and Brownian bridges. In Part I of the series we…
We present a direct bijection between planar 3-connected triangulations and bridgeless planar maps, which were first enumerated by Tutte (1962) and Walsh and Lehman (1975) respectively. Previously known bijections by Wormald (1980) and Fusy…
The Jacobian group ${\rm Jac}(G)$ of a finite graph $G$ is a group whose cardinality is the number of spanning trees of $G$. $G$ also has a tropical Jacobian which has the structure of a real torus; using the notion of break divisors, An et…
Symmetric Dyck tilings and ballot tilings are certain tilings in the region surrounded by two ballot paths. We study the relations of combinatorial objects which are bijective to symmetric Dyck tilings such as labeled trees, Hermite…
This paper deals with the distribution of descent number in standard Young tableaux of certain shapes. A simple explicit formula is presented for the number of tableaux of any shape with two rows, with any specified number of descents. For…
Given a coprime pair $(m,n)$ of positive integers, rational Catalan numbers $\frac{1}{m+n} \binom{m+n}{m,n}$ counts two combinatorial objects:rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the…
We introduce lecture hall tableaux, which are fillings of a skew Young diagram satisfying certain conditions. Lecture hall tableaux generalize both lecture hall partitions and anti-lecture hall compositions, and also contain reverse…
For $0\leq k\leq n-1$, we introduce a family of $k$-skeletal paths which are counted by the $n$-th Catalan number for each $k$, and specialize to Dyck paths when $k=n-1$. We similarly introduce $k$-skeletal parking functions which are…
We consider the decomposition into irreducible components of the external power $\Lambda^p(\mathbb{C}^m\otimes \mathbb{C}^n)$ regarded as a $\operatorname{GL}_m\times\operatorname{GL}_n$-module. Skew Howe duality implies that the Young…
In this paper, we give bijections between the set of 4123-avoiding down-up alternating permutations of length $2n$ and the set of standard Young tableaux of shape $(n,n,n)$, and between the set of 4123-avoiding down-up alternating…
We show that the combinatorial types of the links of the vertices in the edgewise triangulation $T_{k,q}$ of a $(k-1)$-simplex are encoded by the partitions of $k$. Each of these complexes is isomorphic to a subcomplex of the barycentric…
Our main contribution here is the discovery of a new family of standard Young tableaux $ {\cal T}^k_n$ which are in bijection with the family ${\cal D}_{m,n}$ of Rational Dyck paths for $m=k\times n\pm 1$ (the so called "Fuss" case). Using…
We consider decompositions of digraphs into edge-disjoint paths and describe their connection with the $n$-th Weyl algebra of differential operators. This approach gives a graph-theoretic combinatorial view of the normal ordering problem…
We study the geometry of a birational map between an intersection of a web of quadrics in seven-dimensional complex projective space that contains a plane and the double octic branched along the discriminant of the web.
We consider two questions of Wilf related to Standard Young Tableaux. We provide a partial answer to one question, and that will lead us to a more general answer to the other question. Our answers are purely combinatorial.
In this paper, we mainly consider a combinatoric explanation for block Pfaffians in terms of non-intersecting paths, as a generalization of results obtained by Stembridge. As applications, we demonstrate how are generating functions of…
Shi tableaux are special binary fillings of certain Young diagrams which arise in the study of Shi hyperplane arrangements related to classical root systems. For type $A$, the set $\mathcal T$ of Shi tableaux naturally coincides with the…
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…
In 2010, B. Rhoades proved that promotion together with the fake-degree polynomial associated with rectangular standard Young tableaux give an instance of the cyclic sieving phenomenon. We extend this result to all skew standard Young…
The study of representations of affine Hecke algebras has led to a new notion of shapes and standard Young tableaux which works for the root system of any finite Coxeter group. This paper is completely independent of affine Hecke algebra…