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For 80 years, Dyson’s rank has been known as the partition statistic that witnesses the first two of Ramanujan’s celebrated congruences for the ordinary partition function. In this paper, we show that Dyson’s rank actually witnesses families of partition congruences modulo every prime ℓ \ell . This comes from an in-depth study of when a “multiplicity-based statistic” is a crank witnessing congruences for the function p ( n , m ) p\big (n,m\big ) , which enumerates partitions of n n with parts of size at most m m . We also show that as the modulus ℓ \ell increases, there is an ever-growing collection of distinct multiplicity-based cranks witnessing these same families.

In two papers, Little and Sellers introduced an exciting new combinatorial method for proving partition identities which is not directly bijective. Instead, they consider various sets of weighted tilings of a $$1 \times \infty $$ board with squares and dominoes, and for each type of tiling they construct a generating function in two different ways, which in turn generates a q-series identity. Using this method, they recover quite a few classical q-series identities, but Euler’s Pentagonal Number Theorem is not among them. In this paper, we introduce a key parameter when constructing the generating functions of various sets of tilings which allows us to recover Euler’s Pentagonal Number Theorem along with an uncountably infinite family of generalizations.

We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the Identity Theorem) to prove the full version. We prove three somewhat unusual summation formulae, and use these to give hybrid proofs of a number of identities due to Ramanujan. Finally, we use these new summation formulae to give new partition interpretations of the Rogers-Ramanujan identities and the Rogers-Selberg identities., 20 pages

Andrews defined $$\textit{spt}(n)$$ to be the total number of appearances of the smallest parts in all of the partitions of $$n$$ . In this paper, we study the statistical distribution of $$\textit{spt}(\pi )$$ , the number of smallest parts in the partition $$\pi $$ as $$\pi $$ ranges over all partitions of $$n$$ . We also give a combinatorial proof of a conjecture of Hirschhorn, namely that $$\begin{aligned} p(0)+\ \cdots \ +p(n-1)1$$ , where $$p(n)$$ is the number of partitions of $$n$$ .

If A is a set of nonnegative integers containing 0, then there is a unique nonempty set B of nonnegative integers such that every positive integer can be written in the form a + b, where a ∈ A and b ∈ B, in an even number of ways. We compute the natural density of B for several specific sets A, including the Prouhet–Thue–Morse sequence, {0} ∪ {2n:n ∈ ℕ}, and random sets, and we also study the distribution of densities of B for finite sets A. This problem is motivated by Euler's observation that if A is the set of n that has an odd number of partitions, then B is the set of pentagonal numbers {n(3n + 1)/2:n ∈ ℤ}. We also elaborate the connection between this problem and the theory of de Bruijn sequences and linear shift registers.

In 1994, the following infinite family of congruences was conjectured for the partition functioncΦ2(n)which counts the number of2-colored Frobenius partitions ofn: for alln≥0andα≥1,cΦ2(5αn+λα)≡0(mod5α), whereλαis the least positive reciprocal of12modulo5α. In this paper, the first four cases of this family are proved.

The ordinary partition function p(n) and some of its generalizations satisfy some beautiful congruence properties. For instance, Ramanujan proved that for every integer n $$\begin{array}{*{20}c} {p(5n + 4) \equiv 0\,(\bmod \,5),} \\ {p(7n + 5) \equiv 0\,(\bmod \,7),} \\ {p(11n + 6) \equiv 0\,(\bmod \,11).} \\ \end{array}$$ Here we consider congruences for p(n) and c h (n), the number of partitions of n into h colors. If l is prime and s is a positive integer, then, using a result of Sturm, we compute a constant C(h, t, r, l S ) such that c h (tn + r) ≡ 0 (mod l s ) for all n if and only if the congruence holds for every n ≤ C(h, t, r, l s ). If h = 1, these results pertain to p(n). In many cases, C(h, t, r, l S ) is small enough that one obtains an effective method of determining the truth of alleged congruences. For example, Ramanujan’s congruences are easily verified because C(1, 5, 4, 5) = 2, C(1, 7, 5, 7) = 4, and C(1, 11,6,11) = 10.

Recently, several authors have been treating partitions by considering the gaps between their consecutive parts. Many of the existing results involving “partitions with parts in the gaps” bear a striking resemblance to certain weighted partition identities that also arise from the gaps of partitions. In this paper, we show that this is not merely a coincidence, and we demonstrate that many of these weighted partition identities are equivalent to identities involving partitions with parts in the gaps. We intend that this brief note will more or less act as a Rosetta Stone for some of the literature on the gap-theoretic study of partitions.

The parity of p(n), the ordinary partition function, has been studied for at least a century, yet it still remains something of a mystery. Although much work has been done, the known lower bounds for the number of even and odd values of p(n) for n ≤ N still appear to have a great deal of room for improvement. In this paper, we use classical methods to give a new lower bound for the number of odd values of p(n).