scholarly journals Generalized Catalan Numbers from Hypergraphs

10.37236/8733 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Paul E. Gunnells
Keyword(s):  
Matrix Models ◽  
Associative Algebra ◽  
Matrix Model ◽  
Catalan Numbers ◽  
Set Partitions ◽  
Generalized Catalan Numbers ◽  
Combinatorial Interpretations ◽  
Plane Trees ◽  
Infinite Collection

The Catalan numbers $C_{n} \in \{1,1,2,5,14,42,\dots \}$ form one of the most venerable sequences in combinatorics. They have many combinatorial interpretations, from counting bracketings of products in non-associative algebra to counting rooted plane trees and noncrossing set partitions. They also arise in the GUE matrix model as the leading coefficient of certain polynomials, a connection closely related to the plane trees and noncrossing set partitions interpretations. In this paper we define a generalization of the Catalan numbers. In fact we actually define an infinite collection of generalizations $C_{n}^{(m)}$, $m\geq 1$, with $C_{n}^{(1)}$ equal to the usual Catalans $C_{n}$; the sequence $C_{n}^{(m)}$ comes from studying certain matrix models attached to hypergraphs. We also give some combinatorial interpretations of these numbers.

scholarly journals The Delta Conjecture

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
James Haglund ◽  
Jeffrey B. Remmel ◽  
Andrew Timothy Wilson
Keyword(s):  
Symmetric Function ◽  
Ordered Set ◽  
Macdonald Polynomials ◽  
Catalan Numbers ◽  
Set Partitions ◽  
Combinatorial Interpretations ◽  
Llt Polynomials ◽  
Ordered Set Partitions ◽  
International Audience ◽  
Tesler Matrices

International audience We conjecture two combinatorial interpretations for the symmetric function ∆eken, where ∆f is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture, a statement originally conjectured by Haglund, Haiman, Remmel, Loehr, and Ulyanov and recently proved by Carlsson and Mellit. We show how previous work of the second and third authors on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to prove another case. Finally, we show how our conjectures inspire 4-variable generalizations of the Catalan numbers, extending work of Garsia, Haiman, and the first author.

scholarly journals Some Combinatorial Interpretations and Applications of Fuss-Catalan Numbers

2011 ◽  
Vol 2011 ◽  
pp. 1-8 ◽  
Author(s):  
Chin-Hung Lin
Keyword(s):  
Catalan Numbers ◽  
Catalan Number ◽  
Basic Properties ◽  
Generalized Catalan Numbers ◽  
Combinatorial Interpretations

Fuss-Catalan number is a family of generalized Catalan numbers. We begin by two definitions of Fuss-Catalan numbers and some basic properties. And we give some combinatorial interpretations different from original Catalan numbers. Finally we generalize the Jonah's theorem as its applications.

scholarly journals Genus expansion of matrix models and ћ expansion of KP hierarchy

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
A. Andreev ◽  
A. Popolitov ◽  
A. Sleptsov ◽  
A. Zhabin
Keyword(s):  
Matrix Models ◽  
Matrix Model ◽  
Special Interest ◽  
Hermitian Matrix ◽  
Geometric Meaning ◽  
Kp Hierarchy ◽  
Hurwitz Numbers ◽  
Kp Equations ◽  
Genus Expansion ◽  
Hermitian Matrix Model

Abstract We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

scholarly journals Multiple phases in a generalized Gross-Witten-Wadia matrix model

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Jorge G. Russo ◽  
Miguel Tierz
Keyword(s):  
Partition Function ◽  
Matrix Models ◽  
Characteristic Polynomial ◽  
Matrix Model ◽  
Phase Structure ◽  
Unitary Matrix ◽  
Two Dimensional ◽  
Toeplitz Determinants ◽  
Dimensional Parameter ◽  
Planar Limit

Abstract We study a unitary matrix model of the Gross-Witten-Wadia type, extended with the addition of characteristic polynomial insertions. The model interpolates between solvable unitary matrix models and is the unitary counterpart of a deformed Cauchy ensemble. Exact formulas for the partition function and Wilson loops are given in terms of Toeplitz determinants and minors and large N results are obtained by using Szegö theorem with a Fisher-Hartwig singularity. In the large N (planar) limit with two scaled couplings, the theory exhibits a surprisingly intricate phase structure in the two-dimensional parameter space.

PHASE DIAGRAM AND ORTHOGONAL POLYNOMIALS IN MULTIPLE-WELL MATRIX MODELS

1991 ◽  
Vol 06 (25) ◽  
pp. 4491-4515 ◽  
Author(s):  
OLAF LECHTENFELD ◽  
RASHMI RAY ◽  
ARUP RAY
Keyword(s):  
Phase Diagram ◽  
Orthogonal Polynomials ◽  
Matrix Models ◽  
Matrix Model ◽  
Phase Structure ◽  
Saddle Points ◽  
Tree Level ◽  
Potential Wells ◽  
Large N

We investigate a zero-dimensional Hermitian one-matrix model in a triple-well potential. Its tree-level phase structure is analyzed semiclassically as well as in the framework of orthogonal polynomials. Some multiple-arc eigenvalue distributions in the first method correspond to quasiperiodic large-N behavior of recursion coefficients for the second. We further establish this connection between the two approaches by finding three-arc saddle points from orthogonal polynomials. The latter require a modification for nondegenerate potential minima; we propose weighing the average over potential wells.

scholarly journals On Pattern-Avoiding Partitions

10.37236/763 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Stirling Numbers ◽  
Equivalence Classes ◽  
Catalan Numbers ◽  
Set Partition ◽  
Combinatorial Interpretations

A set partition of size $n$ is a collection of disjoint blocks $B_1,B_2,\ldots$, $B_d$ whose union is the set $[n]=\{1,2,\ldots,n\}$. We choose the ordering of the blocks so that they satisfy $\min B_1 < \min B_2 < \cdots < \min B_d$. We represent such a set partition by a canonical sequence $\pi_1,\pi_2,\ldots,\pi_n$, with $\pi_i=j$ if $i\in B_j$. We say that a partition $\pi$ contains a partition $\sigma$ if the canonical sequence of $\pi$ contains a subsequence that is order-isomorphic to the canonical sequence of $\sigma$. Two partitions $\sigma$ and $\sigma'$ are equivalent, if there is a size-preserving bijection between $\sigma$-avoiding and $\sigma'$-avoiding partitions. We determine all the equivalence classes of partitions of size at most $7$. This extends previous work of Sagan, who described the equivalence classes of partitions of size at most $3$. Our classification is largely based on several new infinite families of pairs of equivalent patterns. For instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. Our results also yield new combinatorial interpretations of the Catalan numbers and the Stirling numbers.

scholarly journals Quantum gravity as a multitrace matrix model

2017 ◽  
Vol 32 (31) ◽  
pp. 1750180
Author(s):  
Badis Ydri ◽  
Cherine Soudani ◽  
Ahlam Rouag
Keyword(s):  
Quantum Gravity ◽  
Matrix Models ◽  
Large Class ◽  
Matrix Model ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Topology Change ◽  
New Model ◽  
And Topology ◽  
Emergent Geometry

We present a new model of quantum gravity as a theory of random geometries given explicitly in terms of a multitrace matrix model. This is a generalization of the usual discretized random surfaces of two-dimensional quantum gravity which works away from two dimensions and captures a large class of spaces admitting a finite spectral triple. These multitrace matrix models sustain emergent geometry as well as growing dimensions and topology change.

β-DEFORMED MATRIX MODELS AND NEKRASOV PARTITION FUNCTION

2013 ◽  
Vol 21 ◽  
pp. 92-100
Author(s):  
TAKESHI OOTA
Keyword(s):  
Gauge Theory ◽  
Partition Function ◽  
Matrix Models ◽  
Matrix Model ◽  
The Matrix

The β-deformed matrix models of Selberg type are introduced. They are exactly calculable by using the Macdonald-Kadell formula. With an appropriate choice of the integration contours and interactions, the partition function of the matrix model can be identified with the Nekrasov partition function for SU(2) gauge theory with Nf = 4. Known properties of good q-expansion basis for the matrix model are summarized.

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