相关论文: Une identit\'e en th\'eorie des partitions

Using previous work by Merca, we show the partition function involving parts of k different magnitudes, shifted by the triangular numbers, equals the self convolution of the unrestricted partition function. We also provide a combinatorial…

We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers-Ramanujan type identity. This means that a single analytical Rogers-Ramanujan type identity implies the existence of…

Using elementary means, we prove several identities involving the M\"obius function, generalizing in the multidimensional case well-known formulas coming from convolution arguments.

We prove a family of partition identities which is "dual" to the family of Andrews-Gordon's identities. These identities are inspired by a correspondence between a special type of partitions and "hypergraphs" and their proof uses…

We obtain a finite form of Jacobi's identity and present a combinatorial proof based on the structure of synchronized partitions.

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by…

We formulate several polynomial identities. One side of these identities has a nice simple form. Whereas the other has a form of a polynomial whose coefficients contain binomial coefficients double factorials or (and) rising factorials. The…

We provide bijective proofs of two classic identities that are very simple to prove using generating functions, but surprisingly difficult to prove combinatorially. The problem of finding a bijective proof for the first identity was first…

We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of…

In this note, we present a curious $q$-series identity with applications to certain partitions with bounded part differences.

We give a new proof of Chan's identity involving the cubic partition function and we also give a new identity for the cubic partition function which is analogues to the Zuckerman's identity for the ordinary partition function.

We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy's Durfee square identity to multipartitions. We…

New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…

In this note, using the derangement polynomials and their umbral representation, we give another simple proof of an identity conjectured by Lacasse in the study of the PAC-Bayesian machine learning theory.

In this brief note a straightforward combinatorial proof for an identity directly connecting rooted forests and unordered set partitions is provided. Furthermore, references that put this type of identity in the context of forest volumes…

In this paper, we give a conjecture, which generalises Euler's partition theorem involving odd parts and different parts for all moduli. We prove this conjecture for two family partitions. We give $q$-difference equations for the related…

We give an elementary probabilistic proof of a binomial identity. The proof is obtained by computing the probability of a certain event in two different ways, yielding two different expressions for the same quantity.

In this note, we provide a conceptual explanation of a well-known polynomial identity used in algebraic number theory.

We show that cylindric partitions are in one-to-one correspondence with a pair which has an ordinary partition and a colored partition into distinct parts. Then, we show the general form of the generating function for cylindric partitions…