The $q,t$-Catalan number $\mathrm{Cat}_n(q,t)$ enumerates integer partitions contained in an $n\times n$ triangle by their dinv and external area statistics. The paper [LLL18 (Lee, Li, Loehr, SIAM J. Discrete Math. 32(2018))] proposed a new approach to understanding the symmetry property $\mathrm{Cat}_n(q,t)=\mathrm{Cat}_n(t,q)$ based on decomposing the set of all integer partitions into infinite chains. Each such global chain $\mathcal{C}_μ$ has an opposite chain $\mathcal{C}_{μ^*}$; these combine to give a new small slice of $\mathrm{Cat}_n(q,t)$ that is symmetric in $q$ and $t$. Here we advance the agenda of [LLL18] by developing a new general method for building the global chains $\mathcal{C}_μ$ from smaller elements called local chains. We define a local opposite property for local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most $11$. This proves that for all $n$, the terms in $\mathrm{Cat}_n(q,t)$ of degree at least $\binom{n}{2}-11$ are symmetric in $q$ and $t$.
Following Benjamin et al., a matrix with entries being sums of two neighbouring Catalan numbers is considered. Its LU-decomposition is given, by guessing the results and later prove it by computer algebra, with lots of human help. Specializing a parameter, the determinant turns out to be a Fibonacci number with odd index, confirming earlier results, obtained back then by combinatorial methods.
A positroid is the matroid of a matrix whose maximal minors are all nonnegative. Given a permutation $w$ in $S_n$, the matroid of a generic $n \times n$ matrix whose non-zero entries in row $i$ lie in columns $w(i)$ through $n+i$ is an example of a positroid. We enumerate the bases of such a positroid as a sum of certain products of Catalan numbers, each term indexed by the $123$-avoiding permutations above $w$ in Bruhat order. We also give a similar sum formula for their Tutte polynomials. These are both avatars of a structural result writing such a positroid as a disjoint union of matroids, …
Analysis of the dynamics of the Dyck words helped solve the problem of representing the Catalan number as a sum of squares of natural numbers. In this case, the Dyck triangle is considered in different coordinates. In the calculations, we use the Catalan convolution matrix.
While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and conjectures) are presented on this splitting of the squares that contribute to a refined understanding of the Kronecker squares. Furthermore, connections to 2-modular decomposition numbers, Catalan combinatorics, and to the Saxl conjecture are discussed which further motivate the study of these splittings.
Motivated by the Hankel determinant evaluation of moment sequences, we study a kind of Pfaffian analogue evaluation. We prove an LU-decomposition analogue for skew-symmetric matrices, called Pfaffian decomposition. We then apply this formula to evaluate Pfaffians related to some moment sequences of classical orthogonal polynomials. In particular we obtain a product formula for a kind of q-Catalan Pfaffians. We also establish a connection between our Pfaffian formulas and certain weighted enumeration of shifted reverse plane partitions.
This article is part of an ongoing investigation of the combinatorics of $q,t$-Catalan numbers $\textrm{Cat}_n(q,t)$. We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle height. Our goal is to decompose the infinite set of partitions of deficit $k$ into a disjoint union of chains $\mathcal{C}_μ$ indexed by partitions of size $k$. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property $\textrm{Cat}_n(q,t)=\textrm{Cat}_n(t,q)$. Previously, we introduced a map that bui…
We consider the Cauchy problem for the Volterra chain with an initial condition equal to 0 in one node and 1 in the others. It is shown that this problem admits an exact solution in terms of the Bessel functions. The Taylor series arising here are related to the exponential generating function for Catalan numbers.
The higher $q,t$-Catalan polynomial $C^{(m)}_n(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of $n$. This paper proves the equivalence of the two definitions for all $m\geq 1$ and all $n\leq 4$. We also give a bijective proof of the joint symmetry property $C^{(m)}_n(q,t)=C^{(m)}_n(t,q)$ for all $m\geq 1$ and all $n\leq 4$. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes…
Catalan numbers arise in many enumerative contexts as the counting sequence of combinatorial structures. In this work, we consider natural Markov chains on some of the realizations of the Catalan sequence. While our main result is in deriving an $O(n^2 \log n)$ bound on the mixing time in $L_2$ (and hence total variation) distance for the random transposition chain on Dyck paths, we raise several open questions, including the optimality of the above bound. The novelty in our proof is in establishing a certain negative correlation property among random bases of lattice path matroids, including …
Recent work of the first author, Negut and Rasmussen, and of Oblomkov and Rozansky in the context of Khovanov--Rozansky knot homology produces a family of polynomials in $q$ and $t$ labeled by integer sequences. These polynomials can be expressed as equivariant Euler characteristics of certain line bundles on flag Hilbert schemes. The $q,t$-Catalan numbers and their rational analogues are special cases of this construction. In this paper, we give a purely combinatorial treatment of these polynomials and show that in many cases they have nonnegative integer coefficients. For sequences of length…
The classic Dyck triangle, the Catalan triangle, and the Catalan convolution matrix are plane projections of the multidimensional Dyck triangle. In the Dyck path, each node is uniquely determined by two of four interrelated parameters: (i) the position of the current parenthesis, (ii) the current unbalance of the parentheses, (iii) the number of viewed left parentheses, and (iv) the same for right parentheses. The last two parameters can be redefined, respectively, as the index of the current Catalan number and the index of the summand in the decomposition of the Catalan number into the sum of…
The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for $C_n(q,t)$ as weighted sums of Dyck paths (or equivalently, integer partitions contained in a staircase shape). This paper undertakes a combinatorial investigation of the joint symmetry property $C_n(q,t)=C_n(t,q)$. We conjecture some structural decompositions of Dyck objects into "mutually op…
The d-dimensional Catalan numbers form a well-known sequence of numbers which count balanced bracket expressions over an alphabet of size d. In this paper, we introduce and study what we call d-dimensional prime Catalan numbers, a sequence of numbers which count only a very specific subset of indecomposable balanced bracket expressions. We further introduce the notion of a trapezoidal diagram of a crossing-free geometric graph, such as a triangulation or a crossing-free perfect matching. In essence, such a diagram is obtained by augmenting the geometric graph in question with its trapezoidal d…
It is well-known that the triangulations of the disc with $n+2$ vertices on its boundary are counted by the $n$th Catalan number $C(n)=\frac{1}{n+1}{2n \choose n}$. This paper deals with the generalisation of this problem to any arbitrary compact surface $S$ with boundaries. We obtain the asymptotic number of simplicial decompositions of the surface $S$ with $n$ vertices on its boundary. More generally, we determine the asymptotic number of dissections of $S$ when the faces are $δ$-gons with $δ$ belonging to a set of admissible degrees $Δ\subseteq \{3,4,5,...\}$. We also give the limit laws of…
Towards the study of the Kashiwara B(infinity) crystal, sets H^t of functions were introduced given by equivalence classes of unordered partitions satisfying certain boundary conditions. Here it is shown that H^t is a Catalan set of order t, that is to say the cardinality of H^t is the t-th Catalan number C(t). This is a new description of a Catalan set and moreover admits some remarkable features. Thus to H^t there is an associated labelled graph G_t which is shown to have a canonical decomposition into (t-1)! subgraphs each with 2^{t-1} vertices. These subgraphs, called S-graphs, have some t…
Catalan numbers, parking functions, permutahedra and noncommutative Hilbert schemes
Indiana University（印第安纳大学）；National Research University Higher School of Economics（国立研究大学高等经济大学）；Université Libre de Bruxelles（布鲁塞尔自由大学）；Universiteit Hasselt（哈瑟尔特大学）；Vrije Universiteit Brussel（布鲁塞尔自由大学）
In 1979, Shearer and Kleitman conjectured that there exist $\lfloor n/2 \rfloor+1$ orthogonal chain decompositions of the hypercube $Q_n$, and constructed two orthogonal chain decompositions. In this paper, we make the first non-trivial progress on this conjecture since by constructing three orthogonal chain decompositions of $Q_n$ for $n$ large enough. To do this, we introduce the notion of "almost orthogonal symmetric chain decompositions". We explicitly describe three such decompositions of $Q_5$ and $Q_7$, and describe conditions which allow us to decompose products of hypercubes into $k$ …
We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is "taut", i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \ldots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions --- the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symme…
This is a survey on the categorification of the poset of generalized non-crossing partitions, using the representation theory of a hereditary artin algebra H, looking at the set P of exceptional subcategories in mod H. This categorification is due to Ingalls and Thomas, and a subsequent paper by Igusa and Schiffler. Starting point is a refinement of the classical tilting theory for mod H, replacing torsion pairs by torsion triples, thus putting it into the realm of the stability theory of King. The torsion pairs in mod H correspond nicely to the perpendicular pairs of exceptional subcategories…
We review the Green/Kleitman/Leeb interpretation of de Bruijn's symmetric chain decomposition of ${\cal B}_{n}$, and explain how it can be used to find a maximal collection of disjoint symmetric chains in the nonsymmetric lattice of partitions of a set.
We give a new Jacobi--Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identitie…
Let $Φ$ be an irreducible crystallographic root system with Weyl group $W$ and coroot lattice $\check{Q}$, spanning a Euclidean space $V$. Let $m$ be a positive integer and $\aA^m_Φ$ be the arrangement of hyperplanes in $V$ of the form $(α, x) = k$ for $α\in Φ$ and $k = 0, 1,...,m$. It is known that the number $N^+ (Φ, m)$ of bounded dominant regions of $\aA^m_Φ$ is equal to the number of facets of the positive part $Δ^m_+ (Φ)$ of the generalized cluster complex associated to the pair $(Φ, m)$ by S. Fomin and N. Reading. We define a statistic on the set of bounded dominant regions of $\aA^m_Φ$…
Subgroups of the symmetric group $S_n$ act on powers of chains $C^n$ by permuting coordinates, and induce automorphisms of the ordered sets $C^n$. The quotients defined are candidates for symmetric chain decompositions. We establish this for some families of groups in order to enlarge the collection of subgroups $G$ of the symmetric group $S_n$ for which the quotient $B_n/G$ obtained from the $G$-orbits on the Boolean lattice $B_n$ is a symmetric chain order. The methods are also used to provide an elementary proof that quotients of powers of SCOs by cyclic groups are SCOs.
The $m$-Tamari lattices $\mathcal{T}_{n}^{(m)}$, introduced by Bergeron and Pr{é}ville-Ratelle, are defined as a poset of $m$-Dyck paths equipped with the generalized rotation order, and constitute a Fuss-Catalan generalization of the classical Tamari lattices $\mathcal{T}_{n}$. While for $\mathcal{T}_{n}$ many combinatorial realizations are known, to present there is no further combinatorial realization of $\mathcal{T}_{n}^{(m)}$. In this article, we introduce a certain decomposition of $m$-Dyck paths into $m$-tuples of Dyck paths, and after a certain modification of these $m$-tuples, we conj…
A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a ``strong'' way, is proved. The result is motivated by a conjecture of Graham's concerning probability correlation inequalities for linear extensions of finite posets.
We create a framework for studying symmetric chain decompositions of families of finite posets based on the geometry of polytopes. Our framework unifies almost all known results regarding symmetric chain decompositions of the Young posets $L(m,n)$ --- arising as cells in the Bruhat decomposition of quotients of $SL_{m+n+1}$ --- and yields unexpected new results. The methods we provide are geometric in nature, systematic, and totally amenable to human analysis. This allows us to discover new phenomena which are impenetrable to casework and brute force computer search. In particular, our method …
A poset is called a symmetric chain decomposition if the poset can be expressed as a disjoint union of symmetric chains. For positive integers $m$ and $n$, let $N(m,n)$ denote the set of all compositions $α=(α_1,\cdots,α_m)$, with $0\le α_i \le n$ for each $i=1,\cdots,m$. Define order $<$ as follow, $\forall α,β\in N(m,n)$, $β< α$ if and only if $β_i \le α_i(i=1,\cdots,m)$ and $\sum\limits_{i=1}^{m}β_i <\sum\limits_{i=1}^{m}α_i$. In this paper, we show that the poset $(N(m,n),<)$ can be expressed as a disjoint of symmetric chains by constructive method.
Young's lattice $L(m,n)$ consists of partitions having $m$ parts of size at most $n$, ordered by inclusion of the corresponding Ferrers diagrams. K. O'Hara gave the first constructive proof of the unimodality of the Gaussian polynomials by expressing the underlying ranked set of $L(m,n)$ as a disjoint union of products of centered rank-unimodal subsets. We construct a finer decomposition which is compatible with the partial order on Young's lattice, at the cost of replacing the cartesian product with a more general poset extension. As a corollary, we obtain an explicit chain decomposition whic…
We introduce a method of decomposing the family of intervals along a cyclic permutation into chains to determine the size of the largest family of subsets of $[n]:= \{1,2,...,n\}$ not containing one or more given posets as a subposet. De Bonis, Katona and Swanepoel determined the size of the largest butterfly-free family. We strengthen this result by showing that, for certain posets containing the butterfly poset as a subposet, the same bound holds. We also obtain the corresponding LYM-type inequalities.
The Boolean lattice $2^{[n]}$ is the family of all subsets of $[n]=\{1,\dots,n\}$ ordered by inclusion, and a chain is a family of pairwise comparable elements of $2^{[n]}$. Let $s=2^{n}/\binom{n}{\lfloor n/2\rfloor}$, which is the average size of a chain in a minimal chain decomposition of $2^{[n]}$. We prove that $2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains such that all but at most $o(1)$ proportion of the chains have size $s(1+o(1))$. This asymptotically proves a conjecture of Füredi from 1985. Our proof is based on probabilistic arguments. To analyze our random…
The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion. A chain $c_{0}\subset...\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if $|c_{i}|+|c_{k-i}|=n$ for $i=0,...,k$; and it is symmetric, if $|c_{i}|=(n-k)/2+i$. We show that there exist a bijection $$p: [n]^{(\geq n/2)}\rightarrow [n]^{(\leq n/2)}$$ and a partial ordering $<$ on $[n]^{(\geq n/2)}$ satisfying the following properties: (i) $\subset$ is an extension of $<$ on $[n]^{(\geq n/2)}$; (ii) if $C\subset [n]^{(\geq n/2)}$ is a chain with respect to $<$, then $p(C)\cup C$ is a rank-symmetric chain in $2^{[n]}$, whe…
There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, an…
The $n$-cube is the poset obtained by ordering all subsets of $\{1,\ldots,n\}$ by inclusion, and it can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains, which is the minimum possible number. Two such decompositions of the $n$-cube are called orthogonal if any two chains of the decompositions share at most a single element. Shearer and Kleitman conjectured in 1979 that the $n$-cube has $\lfloor n/2\rfloor+1$ pairwise orthogonal decompositions into the minimum number of chains, and they constructed two such decompositions. Spink recently improved this by showing that the $n$-cube has …
A finite ranked poset is called a symmetric chain order if it can be written as a disjoint union of rank-symmetric, saturated chains. If $P$ is any symmetric chain order, we prove that $P^n/\mathbb{Z}_n$ is also a symmetric chain order, where $\mathbb{Z}_n$ acts on $P^n$ by cyclic permutation of the factors.
We enumerate the number of monotonic lattice paths starting at $(0,0)$ and terminating at $(m,n)$ in which $l$ of the first $k$ steps lie below the line $y=x\ (0\leq k\leq m\leq n)$. These closed formulas consist of terms which are a product Catalan numbers, ballot numbers and binomial coefficients. We then apply the combinatorial formulas to failure analysis by deriving a probability distribution that compares the performance of a $k$-out-of-$m$ system to a $k$-out-of-$n$ system of continuous, independent, and identically distributed random variables. Lastly, we provide asymptotics in a few s…
The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the $\mathfrak{sl}_2$ lowering operator in the theory of crystal bases.
A joint degree matrix (JDM) specifies the number of connections between nodes of given degrees in a graph, for all degree pairs and uniquely determines the degree sequence of the graph. We consider the space of all balanced realizations of an arbitrary JDM, realizations in which the links between any two degree groups are placed as uniformly as possible. We prove that a swap Markov Chain Monte Carlo (MCMC) algorithm in the space of all balanced realizations of an {\em arbitrary} graphical JDM mixes rapidly, i.e., the relaxation time of the chain is bounded from above by a polynomial in the num…
Let H be a subgroup of a finite group G. We use Markov chains to quantify how large r should be so that the decomposition of the r tensor power of the representation of G on cosets on H behaves (after renormalization) like the regular representation of G. For the case where G is a symmetric group and H a parabolic subgroup, we find that this question is precisely equivalent to the question of how large r should be so that r iterations of a shuffling method randomize the Robinson-Schensted-Knuth shape of a permutation. This equivalence is rather remarkable, if only because the representation th…
We consider the problem of constructing a cyclic listing of all bitstrings of length $2n+1$ with Hamming weights in the interval $[n+1-\ell,n+\ell]$, where $1\leq \ell\leq n+1$, by flipping a single bit in each step. This is a far-ranging generalization of the well-known middle two levels problem (the case $\ell=1$). We provide a solution for the case $\ell=2$, and we solve a relaxed version of the problem for general values of $\ell$, by constructing cycle factors for those instances. The proof of the first result uses the lexical matchings introduced by Kierstead and Trotter, which we genera…
Reiner and Webb compute the $S_n$-module structure for the complex of injective words in [RW]. This paper refines their formula by providing a Hodge type decomposition. Along the way, this paper proves that the simplicial boundary map interacts in a nice fashion with the Eulerian idempotents. The Laplacian acting on the top chain group in the complex of injective words is also shown to equal the signed random to random shuffle operator. Uyemura-Reyes conjectures in [Uy] that the (unsigned) random to random shuffle operator has integral spectrum. We prove that this conjecture would imply that t…
Young's partition lattice $L(m,n)$ consists of unordered partitions having $m$ parts where each part is at most $n$. Using methods from complex algebraic geometry, R. Stanley proved that $L(m,n)$ is rank-symmetric, unimodal, and strongly Sperner. Moreover, he conjectured that $L(m,n)$ has a stronger property called symmetric chain decomposition. Despite many efforts, this conjecture has only been proved for $\min(m,n)\leq 4$. In this paper, we decompose $L(m,n)$ into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, …
