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The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of $t$ -cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function...

The number of 0-1-2 increasing trees as two different evaluations of the Tutte polynomial of a complete graph.

Merino, C. (2008)

The Electronic Journal of Combinatorics [electronic only]

The number of complete exceptional sequences for a Dynkin algebra

Mustafa Obaid, Khalid Nauman, Wafa S. M. Al-Shammakh, Wafaa Fakieh, Claus Michael Ringel (2013)

Colloquium Mathematicae

The Dynkin algebras are the hereditary artin algebras of finite representation type. The paper determines the number of complete exceptional sequences for any Dynkin algebra. Since the complete exceptional sequences for a Dynkin algebra of Dynkin type Δ correspond bijectively to the maximal chains in the lattice of non-crossing partitions of type Δ, the calculations presented here may also be considered as a categorification of the corresponding result for non-crossing partitions.

The number-theoretic content of the Jacobi triple product identity.

Wilf, Herbert S. (1999)

Séminaire Lotharingien de Combinatoire [electronic only]

The $q$ -binomial theorem and two symmetric $q$ -identities.

Guo, Victor J.W. (2003)

The Electronic Journal of Combinatorics [electronic only]

The spt-crank for overpartitions

Frank G. Garvan, Chris Jennings-Shaffer (2014)

Acta Arithmetica

Bringmann, Lovejoy, and Osburn (2009, 2010) showed that the generating functions of the spt-overpartition functions $\bar{s p t} (n)$ , ${\bar{s p t}}_{1} (n)$ , ${\bar{s p t}}_{2} (n)$ , and M2spt(n) are quasimock theta functions, and satisfy a number of simple Ramanujan-like congruences. Andrews, Garvan, and Liang (2012) defined an spt-crank in terms of weighted vector partitions which combinatorially explain simple congruences modulo 5 and 7 for spt(n). Chen, Ji, and Zang (2013) were able to define this spt-crank in terms of ordinary partitions. In this...

The structure of standard modules, I: Universal algebras and the Rogers-Ramanujan identities.

James Lepowsky, Robert L. Wilson (1984)

Inventiones mathematicae

The tangent function and power residues modulo primes

Zhi-Wei Sun (2023)

Czechoslovak Mathematical Journal

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$ . When $m$ is a positive integer with $p \equiv 1 (mod 2 m)$ and $2$ is an $m$ th power residue modulo $p$ , we determine the value of the product $\prod_{k \in R_{m} (p)} (1 + tan (π a k / p))$ , where $R_{m} (p) = {0 < k < p : k \in ℤ is an m th power residue modulo p} .$ In particular, if $p = x^{2} + 64 y^{2}$ with $x, y \in ℤ$ , then $\prod_{k \in R_{4} (p)} (1 + tan π \frac{a k}{p}) = {(- 1)}^{y} {(- 2)}^{(p - 1) / 8} .$

The triple, quintuple and setuple product identities revisited.

Foata, Dominique, Han, Guo-Niu (1999)

Séminaire Lotharingien de Combinatoire [electronic only]

The two-parameter class of Schröder inversions

J. Schröder (2013)

Commentationes Mathematicae Universitatis Carolinae

Infinite lower triangular matrices of generalized Schröder numbers are used to construct a two-parameter class of invertible sequence transformations. Their inverses are given by triangular matrices of coordination numbers. The two-parameter class of Schröder transformations is merged into a one-parameter class of stretched Riordan arrays, the left-inverses of which consist of matrices of crystal ball numbers. Schröder and inverse Schröder transforms of important sequences are calculated.

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Tableaux de Young et fonctions de Schur-Littlewood

The 99th Fibonacci identity.

The combinatorialization of linear recurrences.

The combinatorics of the Garsia-Haiman modules for hook shapes.

The concept of Bailey chains.

The finite Heine transformation and conjugate Durfee squares.

The Lagrange inversion formula and divisibility properties.

The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications