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We compute the partition and correlation generating functions for the Heisenberg intertwiner generalized vertex operator algebra on a genus $g$ Riemann surface in the Schottky uniformization. These are expressed in terms of differential forms of the first, second and third kind, the prime form and the period matrix and are computed by combinatorial methods using a generalization of the MacMahon Master Theorem., Comment: 21 pages. To appear in Proceedings of the Dubrovnik conference Representation Theory XVI, published in the AMS Contemporary Mathematics book series

In 1742, Euler found the generating function for P(n). Hardy said Ramanujan was the first, and upto now the only, Mathematician to discover any such properties of P(n). In 1952, Macmahon established a table of P(n) for the first 200 values of n. This paper showed how to find the number of partition of n by using Macmahon’s table. In 1742, Euler also stated the series in the enumeration of partitions. This Paper showed how to generate the Euler’s use of series in the enumeration of partitions. In 1952, Macmahon also quoted the self-conjuate partitions of n. In this Paper, Macmahon’s self-conjugate partitions are explained with the help of array of dots. This paper showed how to prove the Euler’s Theorems with the help of Euler’s device of the introduction of a second parameter z, and showed how to prove the Theorem 3 with the help of Euler’s generating function for P(n), and also showed how to prove the Theorem 4 with the help of certain conditions of P(n).

Trace monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy remarkable algebraic properties such as the existence and unicity of a prime factorization. Because of this, the set of hikes partially ordered by divisibility hosts a plethora of relations in direct correspondence with those found in number theory. Some applications of these results are presented, including an immanantal extension to MacMahon's master theorem and a derivation of the Ihara zeta function from an abelianization procedure.

The normalizer $N_W(W_J)$ of a standard parabolic subgroup $W_J$ of a finite Coxeter group $W$ splits over the parabolic subgroup with complement $N_J$ consisting of certain minimal length coset representatives of $W_J$ in $W$. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type $D_n$) the centralizer $C_W(w)$ of an element $w \in W$ is in a similar way a semidirect product of the centralizer of $w$ in a suitable small parabolic subgroup $W_J$ with complement isomorphic to the normalizer complement $N_J$., final version, 18 pages, to appear in J. Group Theory

By using MacMahon partition analysis technique, the Poincare series for the algebras of invariants of the ternary and quaternary forms of small degrees are calculated.

One of MacMahon's partition theorems says that the number of partitions of n into parts divisible by 2 or 3 equals the number of partitions of n into parts with multiplicity larger than 1. Recently, Holroyd has obtained a generalization. In this short note, we provide a bijective proof of his theorem.

We present a new algebraic extension of the classical MacMahon Master Theorem. The basis of our extension is the Koszul duality for non-quadratic algebras defined by Berger. Combinatorial implications are also discussed., 8 pages

MacMahon developed partition analysis as a calculational and analytic method to produce the generating function for plane partitions. His efforts did not turn out as he had hoped, and he had to spend nearly twenty years finding an alternative treatment. This paper provides a detailed account of our retrieval of MacMahon’s original project. One of the key results obtained with partition analysis is an extension of a theorem of Gansner which generalizes Stanley’s famous trace theorem. This is the twelfth paper in this series on MacMahon’s partition analysis. It has been our belief from the beginning that MacMahon’s ideas could be best exploited by computer implementation, and that was the genesis of our partition analysis project. Our algorithmic version of MacMahon’s method has been implemented in the form of the Mathematica package Omega which is freely available via the web; see [20]. In the back of our minds was always MacMahon’s melodramatic experience with his own invention. He created partition analysis solely to treat the generating functions associated with various classes of plane partitions. Plane partitions are two-dimensional arrays of non-negative integers which are weakly decreasing in rows and columns; for a formal definition, including the notion of k-trace (1.3), see below. This specific project failed, and in this paper we shall retrieve MacMahon’s original project and obtain, using only partition analysis, an extension, namely Theorem 5.4, of a general plane partition theorem originally due to Gansner [13, Theorem 4.2]. For further remarks on plane partition history, in particular, on how partition analysis has led us to a rediscovery and to an alternative proof of Gansner’s theorem, we refer the interested reader to [4]. The initial stage of MacMahon’s investigations is chronicled by him in [16], where he refines the study of partitions and compositions of multipartite numbers into the theory of plane partitions. MacMahon [16, p. 658] first stated as an unproven assertion that the generating function for plane partitions is, in fact

In two previous papers, the study of partitions with short sequences has been developed both for its intrinsic interest and for a variety of applications. The object of this paper is to extend that study in various ways. First, the relationship of partitions with no consecutive integers to a theorem of MacMahon and mock theta functions is explored independently. Secondly, we derive in a succinct manner a relevant definite integral related to the asymptotic enumeration of partitions with short sequences. Finally, we provide the generating function for partitions with no sequences of length K and part exceeding N.

We state and prove a quantum-generalization of MacMahon's celebrated Master Theorem, and relate it to a quantum-generalization of the boson-fermion correspondence of Physics., Comment: AMS-LaTeX, 7 pages, revised and final version. To appear in Proc. Natl. Acad. Sciences

A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. In this article, we give a combinatorial overview of the Hopf algebra structure of the MacMahon symmetric functions relying on the construction of a Hopf algebra from any alphabet of neutral letters obtained in [18, 19].

In his famous book "Combinatory Analysis" MacMahon introduced Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. By developing the Omega package we have shown that Partition Analysis is ideally suited for being supplemented by computer algebra methods. The object of this paper is to present a significant algorithmic improvement of this package. It overcomes a problem related to the computational treatment of roots of unity. Moreover, this new reduction strategy turns out to be superior to "The Method of Elliott" which is described in MacMahon's book. In order to make this article as self-contained as possible we give a brief introduction to Partition Analysis together with a variety of illustrative examples. For instance, the generating function of magic pentagrams is computed.

Using dierected graphs, we present a combinatorial model for the polynomial matrices corresponding to representations of the general linear groups. In doing so, we obtain a very simple combinatorial rule to multiply basic elements of the Schur algebra.

In his famous book ''Combinatory Analysis'' MacMahon introduced Partition Analysis as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. However, MacMahon failed in his attempt to use his method for a satisfactory treatment of plane partitions. It is the object of this article to show that nevertheless Partition Analysis is of significant value when treating non-standard types of plane partitions. To this end ''plane partition diamonds'' are introduced. Applying Partition Analysis a simple closed form for the full generating function is derived. In the discovering process the Omega package developed by the authors has played a fundamental role.

We continue the study of the method outlined by MacMahon in Section VIII of [10]. The long range object is to show the relevance of MacMahon's ideas in current partition-theoretic research. In this paper we present a number of theorems which MacMahon overlooked. For example, the number of partitions of n with non-negative first and second differences between parts equals the number of partitions of n into triangular numbers.

We compute the inverse of a specific infinite r -dimensional matrix, extending a matrix inverse of Krattenthaler. Our inversion is different from the r -dimensional matrix inversion recently found by Schlosser but generalizes a multidimensional matrix inversion previously found by Chu. As applications of our matrix inversion we derive some multidimensional q -series identities. Among these are q -analogues of Carlitz' multidimensional Abel-type expansion formulas. Furthermore, we derive a q -analogue of MacMahon's Master Theorem.

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. As an application, we develop refined Macdonald polynomials for hook shapes. We show that these polynomials are symmetric and give their Schur expansion., Un résultat classique de MacMahon affirme que nombre d’inversion et l’indice majeur ont la même distribution sur permutations d’un multi-ensemble donné. Dans ce travail, nous démontrons un renforcement de ce théorème origine conjecturé par Haglund. Notre résultat peut être considéré comme un théorème d’équirépartition sur les partitions ordonnées d’un multi-ensemble en ensembles, que nous appellerons partitions de multiset commandés. Notre preuve est bijective et implique une nouvelle généralisation de la méthode d’insertion de Carlitz. Comme application, nous développons des polynômes de Macdonald raffinés pour formes d’hameçons. Nous montrons que ces polynômes sont symétriques et donnent leur expansion Schur.

As a unified extension of determinant and permanent, a new matrix function with two extra parameters is introduced. Basic formulas are established, which contain several known relations between determinants and permanents as special cases. Some expansions in terms of the matrix functions are investigated, which yield MacMahon's master theorem, its dual version, and their common generalization.

It is shown that MacMahon's master theorem gives the diagonal elements of a class of irreducible representations of the general linear group,Gl(n,C), and hence the trace of these representations, or the group characters. These representations are unitary under restriction to the unitary subgroup and constitute the so-called totally symmetric representations. A generalization of MacMahon's master theorem that generates the elements of the representation matrix themselves is given. These relations for groups are, in fact, more general in that they apply to arbitrary matrices of indeterminates. These results are proved by using the properties of a class of Young–Weyl tableau polynomials defined overn2indeterminates. Forn=2, two sets of basic polynomials are defined in terms of Young–Weyl tableaux. The first set corresponds to the unitary irreducible representations of U(2), as defined in the physics literature and whose properties comprise a substantive part of what is known as the quantum theory of angular momentum; the second set is Rota's double standard tableau polynomials. These two sets of polynomials span the same space, and a comprehensive set of relations between them is given. A further generalization of MacMahon's master theorem pertaining to the general polynomial representations ofGL(n,C), as enumerated by partitions, is conjectured and is shown to contain the classical Schur function generating function as a special case.

Solutions to a linear Diophantine system, or lattice points in a rational convex polytope, are important concepts in algebraic combinatorics and computational geometry. The enumeration problem is fundamental and has been well studied, because it has many applications in various fields of mathematics. In algebraic combinatorics, MacMahon's partition analysis has become a general approach for linear Diophantine system related problems. Many algorithms have been developed, but "bottlenecks" always arise when dealing with complex problems. While in computational geometry, Barvinok's important result asserts the existence of a polynomial time algorithm when the dimension is fixed. However, the implementation by the LattE package of De Loera et. al. does not perform well in many situations. By combining excellent ideas in the two fields, we generalize Barvinok's result by giving a polynomial time algorithm for MacMahon's partition analysis in a suitable condition. We also present an elementary Euclid style algorithm, which might not be polynomial but is easy to implement and performs well. As applications, we contribute the generating series for magic squares of order 6., 28 pages, some modification, add the link to the Maple package CTEuclid

We consider a number of generalizations of the $\beta$-extended MacMahon Master Theorem for a matrix. The generalizations are based on replacing permutations on multisets formed from matrix indices by partial permutations or derangements over matrix or submatrix indices., Comment: 16 pages, 4 figures

This paper gives a brief exposition of several recent results obtained by the authors concerning some classes of inverse relations, including the general binomial-type inversions and two kinds of general Stirling reciprocal transforms. Also described are some applications of the related inversion techniques to combinatorics (including new proofs of Rogers-Ramanujan identities and MacMahon's master theorem), interpolation methods and certain problems related to special polynomials and number sequences.

The theory of the Mobius function connects the principle of inclusion and exclusion (PIE) with another basic concept, namely the zeta function of a graph. In this section we discuss the theorem of Bass which will be useful in Chapter 9, and the MacMahon Master theorem, which will be useful in Chapter 8. Then we study graph polynomials, which constitute the basic connection between enumeration and the partition functions of statistical physics.

The numerical Hilbert series combinatorics and the comodule Hilbert series combinatorics are introduced, and some applications are presented, including the MacMahon Master Theorem., Comment: 11 pages; to appear in International Journal of Geometric Methods in Mathematical Physics; conference in honour of Michel Dubois-Violette

The right-quantum algebra was introduced recently by Garoufalidis, Lê and Zeilberger in their quantum generalization of the MacMahon master theorem. A bijective proof of this identity due to Konvalinka and Pak, and also the recent proof of the right-quantum Sylvester's determinant identity, make heavy use of a bijection related to the first fundamental transformation on words introduced by Foata. This paper makes explicit the connection between this transformation and right-quantum linear algebra identities; we give a new, bijective proof of the right-quantum matrix inverse theorem, we show that similar techniques prove the right-quantum Jacobi ratio theorem, and we use the matrix inverse formula to find a generalization of the (right-quantum) MacMahon master theorem.

A theorem of Goulden and Jackson which gives interesting formulae for character immanants also implies MacMahon's Master Theorem. We quantize Goulden and Jackson's theorem to give formulae for quantum character immanants in such a way as to obtain a known quantization of MacMahon's Master Theorem due to Garoufalidis-Lê-Zeilberger. In doing so, we also quantize formulae of Littlewood, Merris and Watkins concerning induced character immanants., Le Théorème Master de MacMahon peut être vu comme un corollaire d'un résultat de Goulden et Jackson décrivant certaines formules satisfaites par les caractères immanants. Dans cet article, on obtient une version quantique des formules de Goulden et Jackson, ce qui permet de donner une nouvelle preuve du résultat de Garoufalidis, Lê et Zeilberger sur la quantification du Théorème Master de MacMahon. Ce faisant, on arrive aussi à quantifier les formules de Littlewood, Merris et Watkins concernant les caractères immanants induits.

Journal of Physics A: Mathematical and Theoretical 40, 9179-9200 (2007).; International audience; The box-occupancy distributions arising from weighted rearrangements of a particle system are investigated. In the grand-canonical ensemble, they are characterized by determinantal joint probability generating functions. For doubly non-negative weight matrices, fractional occupancy statistics, generalizing Fermi-Dirac and Bose-Einstein statistics, can be defined. A spatially extended version of these balls-in-boxes problems is investigated.

We propose a new proof of the quantum version of MacMahon's Master Theorem, established by Garoufalidis, Le and Zeilberger., Comment: 8 pages

We generalize the generating formula for plane partitions known as MacMahon's formula as well as its analog for strict plane partitions. We give a 2-parameter generalization of these formulas related to Macdonald's symmetric functions. The formula is especially simple in the Hall-Littlewood case. We also give a bijective proof of the analog of MacMahon's formula for strict plane partitions., Comment: 19 pages, 5 figures

We provide a vertex operator realization for a two-parameter generalization of MacMahon’s formula introduced by M. Vuletić [Trans. Amer. Math. Soc. 361, 2789 (2009)]. Since the generalized MacMahon function is the kernel function of some Macdonald symmetric function, we consider the action of two vertex operators on a state corresponding to a Macdonald symmetric function. It becomes evident that the vertex operators appear to be the creation and annihilation operators, respectively on the state.

We give a new proof of the quantum version of MacMahon's Master Theorem due to Garoufalidis, Le and Zeilberger (one-parameter case) and to Konvalinka and Pak (multiparameter case) by deriving it from known facts about Koszul algebras., Comment: 8 pages; minor changes in the exposition; the multiparameter case (originally due to Konvalinka and Pak) is now covered as well

In his famous book ldquoCombinatory Analysisrdquo MacMahon introduced Partition Analysis (ldquoOmega Calculusrdquo) as a computational method for solving problems in connection with linear diophantine inequalities and equations. The technique has recently been given a new life by G.E. Andrews and his coauthors, who had the idea of marrying it with the tools of to-dayrsquos Computer Algebra. The theory consists of evaluating a certain type of rational function of the form A(lambda)-1 B(1/lambda)-1 by the MacMahon OHgr operator. So far, the case where the two polynomials A and B are factorized as products of polynomials with two terms has been studied in details. In this paper we study the case of arbitrary polynomials A and B. We obtain an algorithm for evaluating the OHgr operator using the coefficients of those polynomials without knowing their roots. Since the program efficiency is a persisting problem in several-variable polynomial Calculus, we did our best to make the algorithm as fast as possible. As an application, we derive new combinatorial identities.

A MacMahon symmetric function is a formal power series in a finite number of alphabets that is invariant under the diagonal action of the symmetric group. We use a combinatorial construction of the different bases of the vector space of MacMahon symmetric functions found by the author to obtain their image under the principal specialization: the powers, rising and falling factorials. Then, we compute the connection coefficients of the different polynomial bases in a combinatorial way.

International audience; An extended version of the classical MacMahon's “Master Theorem” to algebraicstructures much more general than the field of real numbers, namely semi-rings and even pre-semi-rings, is proposed. Due to the fact that many semi-rings or pre-semi-rings are not additive groups, such concepts as the determinant of a matrix cannot be used any more, and the statement of the result itself has to be significantly different. When specialized to the field of real numbers, however, the extended result is shown to reduce to the classical version of MacMahon's “Master Theorem”.

In this paper, we analyze the beautiful theorem of M. Bousquet-Melon and K. Eriksson, the Lecture Hall Partition Theorem, via MacMahon’s Partition Analysis. Their theorem asserts that the number of partitions of n of the form b j + b j−1 +…+ b 1 wherein $$\frac{{b_j }} {j} \geqq \frac{{b_j - 1}} {{j - 1}} \geqq \cdots \geqq \frac{{b_\text{1} }} {\text{1}} \geqq 0$$ equals the number of partitions of n into odd parts each ≦ 2j − 1. As they have noted the theorem reduces to Euler’s classical partition theorem when j → ∞.

P.A. MacMahon developed a calculus for producing generating functions of partitions constrained by linear inequalities relating the various parts of the partitions. Recently we have used MacMahon's method to find new classes of partitions whose generating functions are modular forms (Acta Arith., 126(2007), 281-294), and we have retrieved MacMahon's original project which was to prove his generating function formula for plane partitions. We shall report briefly on these topics and indicate new areas for exploration. In particular, we shall consider the modification of MacMahon's methods in which the inequalities relating the various parts of the partition take into account congruence constraints on the parts.

The relationship between the immanant and the Schur symmetric function is examined. Two expressions for the immanant are given in terms of the determinant. Generalisations include Foata and Zeilberger's β-extension of the MacMahon Master theorem. The relationships to some little known results of Littlewood and to idempotents constructed by Young are given

We introduce the combinatorial notion of heaps of pieces, which gives a geometric interpretation of the Cartier-Foata's commutation monoid. This theory unifies and simplifies many other works in Combinatorics : bijective proofs in matrix algebra (MacMahon Master theorem, inversion matrix formula, Jacobi identity, Cayley-Hamilton theorem), combinatorial theory for general (formal) orthogonal polynomials, reciprocal of Rogers-Ramanujan identities, graph theory (matching and chromatic polynomials). Heaps may bring new light on classical subjects as poset theory. They are related to other fields as Theoretical Computer Science (parallelism) and Statistical Physics (directed animals problem, lattice gas model with hard-core interactions). Complete proofs and definitions are given in sections 2, 3,4,5. Other sections give a summary of possible applications of heaps.

The algebra of linear partial difference operators is investigated, and an elimination procedure demonstrated. Applications to combinatorics are given. In particular, a new proof and a q-analogue of MacMahon’s Master Theorem are given.