Generalized euler-type partition identities

Let A and S be subsets of the natural numbers. Let A′(n) be the number of partitions of n where each part appears exactly m times for some m?A. Let S(n) be the number of partitions of n into parts which are elements of S.     

Parity of the partition function

Although much is known about the partition function, little is known about its parity. For the polynomials D(x):=(Dx²+1)/24, where , we show that there are infinitely many m (resp. n) for which p(D(m)) is even (resp. p(D(n)) is odd) if there is at least one such m (resp. n). We bound the first m and n (if any) in terms of the class number h(−D). For prime D we show that there are indeed infinitely many even values. To this end we construct new modular generating functions using generalized Borcherds products, and we employ Galois representations and locally nilpotent Hecke algebras.     

New analogues of Ramanujan's partition identities

We establish several new analogues of Ramanujan's exact partition identities using the theory of modular functions.     

Some new identities for colored partition

In this paper, we discover several new identities on colored partitions and provide proofs for them. Many colored partition identities presented in the paper do not belong to the general and unified combinatorial framework provided by Sandon and Zanello. Most of our proofs depend upon new modular equations given by employing the method of reciprocation.     

Generalizations of the Bellavitis partition identities

Let p = p(a, b, c) be the number of partitions of a into b parts, no part exceeding c. Bellavitis and perhaps some earlier writers noted that p satisfies three very simple identities. Here p is generalized to a function of k + 1 variables in a natural way. One of the identities then generalizes; the proof of this (which depends on the P. Hall commutator collecting process) is given only for k = 3.     

Bijective proofs of some classical partition identities

A bijective proof of a general partition theorem is given which has as direct corollaries many classical partition theorems due to Euler, Glaisher, Schur, Andrews, Subbarao, and others. It is shown that the bijective proof specializes to give bijective proofs of these classical results and moreover the bijections which result often coincide with bijections which have occurred in the literature. Also given are some sufficient conditions for when two classes of words omitting certain sequences of words are in bijection.     

Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities

The Ramanujan Journal - Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five...     

Variants of classical q-hypergeometric identities and partition implications

Utilizing a six-variable extension of Heine’s q-hypergeometric transformation that we previously obtained, we now derive variants of Heine’s transformation formula and the Lebesgue identity. The variant of Cauchy’s identity also obtained by us earlier is crucial in these derivations. We then establish some new partition identities which are variants of, and shed new light on, some fundamental classical partition identities.     

Ramanujan type identities and congruences for partition pairs

Using elementary methods, we establish several new Ramanujan type identities and congruences for certain pairs of partition functions.     

Parity of sums of partition numbers and squares in arithmetic progressions

In this paper, we prove new infinite families of congruences modulo 2 for broken 11-diamond partitions by using Hecke operators.     

Bijective proofs of partition identities arising from modular equations

We establish generalizations of certain partition theorems originating with modular equations and give bijective proofs for them. As a special case, we give a bijective proof of the Farkas and Kra partition theorem modulo 7.     

Restricted partition functions and identities for degrees of syzygies in numerical semigroups

We derive a set of polynomial and quasipolynomial identities for degrees of syzygies in the Hilbert series of numerical semigroup \(\langle d_1,\ldots ,d_m\rangle \), \(m\ge 2\), generated by an arbitrary set of positive integers \(\left\{ d_1, \ldots ,d_m\right\} \), \(\gcd (d_1,\ldots ,d_m)=1\). These identities are obtained by studying the rational representation of the Hilbert series and the quasipolynomial representation of the Sylvester waves in the restricted partition function. In the cases of symmetric semigroups and complete intersections, these identities become more compact; for the latter we find a simple identity relating the degrees of syzygies with elements of generating set \(\left\{ d_1,\ldots ,d_m\right\} \) and give a new lower bound for the Frobenius number.     

A partition bijection related to the Rogers-Selberg identities and Gordon's theorem

We provide a bijective map from the partitions enumerated by the series side of the Rogers-Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers-Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of Gordon's theorem are also explored.     

Some partition and analytical identities arising from the Alladi,Andrews, Gordon bijection

The Ramanujan Journal - In the work of Alladi et al. (J Algebra 174:636–658, 1995) the authors provided a generalization of the two Capparelli identities involving certain classes of integer...