Partitions weighted by the parity of the crank

The ‘crank’ is a partition statistic which originally arose to give combinatorial interpretations for Ramanujan's famous partition congruences. In this paper, we establish an asymptotic formula and a family of Ramanujan type congruences satisfied by the number of partitions of n with even crank Me(n) minus the number of partitions of n with odd crank Mo(n). We also discuss the combinatorial implications of q-series identities involving Me(n)−Mo(n). Finally, we determine the exact values of Me(n)−Mo(n) in the case of partitions into distinct parts. These values are at most two, and zero for infinitely many n.     

Congruences for generalized Frobenius partitions with 4 colors

We present some congruences involving the functions c?₄(n) and which denote, respectively, the number of generalized Frobenius partitions of n with 4 colors and 4-order generalized Frobenius partitions of n with 4 colors.     

Ranks of partitions modulo 10

In 1954, A.O.L. Atkin and H.P.F. Swinnerton-Dyer established the generating functions for rank differences modulo 5 and 7 for partition functions. In this paper, we derive formulas for the generating functions of ranks of partitions modulo 10 and some inequalities between them.     

The generating functions of the rank and crank modulo 8

Let N(i,m;n) be the number of partitions of n with rank (Dyson) congruent to i (mod m) and let M(j,m;n) be the number of partitions of n with crank (Andrews, Garvan) congruent to j (mod m). I give here the generating functions for the numbers N(i,8;n) and M(j,8;n). I suggest forms for the one hundred power series

q-Analogues of Euler’s Odd = Distinct theorem

Two q-analogues of Euler’s theorem on integer partitions with odd or distinct parts are given. A q-lecture hall theorem is given. Supported by NSF grant DMS-0503660.     

Symmetry distribution between hook length and part length for partitions

It is known that the two statistics on integer partitions “hook length” and “part length” are equidistributed over the set of all partitions of n. We extend this result by proving that the bivariate joint generating function by those two statistics is symmetric. Our method is based on a generating function by a triple statistic much easier to calculate.     

Inequalities for full rank differences of 2-marked Durfee symbols

In this paper, we obtain infinitely many non-trivial identities and inequalities between full rank differences for 2-marked Durfee symbols, a generalization of partitions introduced by Andrews. A certain strict inequality, which almost always holds, shows that identities for Dyson?s rank, similar to those proven by Atkin and Swinnerton-Dyer, are quite rare. By showing an analogous strict inequality, we show that such non-trivial identities are also rare for the full rank, but on the other hand we obtain an infinite family of non-trivial identities, in contrast with the partition theoretic case.     

Simple upper bounds for partition functions

We tweak Siegel’s method to produce a rather simple proof of a new upper bound on the number of partitions of an integer into an exact number of parts. Our approach, which exploits the delightful dilogarithm function, extends easily to numerous other counting functions. This work was supported by PSC/CUNY Research Awards (# 67261-00 36 and # 68327-00 37).     

On the number of partitions into primes

There is, apparently, a persistent belief that in the current state of knowledge it is not possible to obtain an asymptotic formula for the number of partitions of a number n into primes when n is large. In this paper such a formula is obtained. Since the distribution of primes can only be described accurately by the use of the logarithmic integral and a sum over zeros of the Riemann zeta-function one cannot expect the main term to involve only elementary functions. However the formula obtained, when n is replaced by a real variable, is in and is readily seen to be monotonic. Research supported by NSA grant, no. MDA904-03-1-0082.     

New Congruences for the Partition Function

Let p(n) denote the number of unrestricted partitions of a non-negative integer n. In 1919, Ramanujan proved that for every non-negative n Recently, Ono proved for every prime m 5 that there are infinitely many congruences of the form p(An+B)0 (mod m). However, his results are theoretical and do not lead to an effective algorithm for finding such congruences. Here we obtain such an algorithm for primes 13m31 which reveals 76,065 new congruences.     

The solution to the partition reconstruction problem

Given a partition λ of n, a k-minor of λ is a partition of n−k whose Young diagram fits inside that of λ. We find an explicit function g(n) such that any partition of n can be reconstructed from its set of k-minors if and only if k?g(n). In particular, partitions of n?k²+2k are uniquely determined by their sets of k-minors. This result completely solves the partition reconstruction problem and also a special case of the character reconstruction problem for finite groups.     

Divisibility of Certain Partition Functions by Powers of Primes

Let be the prime factorization of a positive integer k and let b _k (n) denote the number of partitions of a non-negative integer n into parts none of which are multiples of k. If M is a positive integer, let S _k (N; M) be the number of positive integers N for which b _k(n ) 0(mod M). If we prove that, for every positive integer j In other words for every positive integer j, b _k(n) is a multiple of for almost every non-negative integer n. In the special case when k=p is prime, then in representation-theoretic terms this means that the number ofp -modular irreducible representations of almost every symmetric groupS _n is a multiple of p ^j. We also examine the behavior of b _k(n) (mod ) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n (mod t) satisfies b _k(n) 0 (mod ), we show that there are infinitely many non-negative integers n r (mod t) for which b _k(n) 0 (mod ) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 .     

ℓ-Divisibility of ℓ-regular partition functions

We give exact criteria for the ℓ-divisibility of the ℓ-regular partition function b _ℓ (n) for ℓ∈{5,7,11}. These criteria are found using the theory of complex multiplication. In each case the first criterion given corresponds to the Ramanujan congruence modulo ℓ for the unrestricted partition function, and the second is a condition given by J.-P. Serre for the vanishing of the coefficients of ∏ _m=1^∞(1−q ^m ) ^ℓ−1.      

Classification of Plane Congruences in 4 C4 (I)

Nondegenerate plane congruences in the four-dimensional complex projective space with degenerate general focal conic are classified by using the focal method due to Corrado Segre.