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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.

In this paper, we use a branch of polyhedral geometry, Ehrhart theory, to expand our combinatorial understanding of congruences for partition functions. Ehrhart theory allows us to give a new decomposition of partitions, which in turn allows us to define statistics called {\it supercranks} that combinatorially witness every instance of divisibility of $p(n,3)$ by any prime $m \equiv -1 \pmod 6$, where $p(n,3)$ is the number of partitions of $n$ into three parts. A rearrangement of lattice points allows us to demonstrate with explicit bijections how to divide these sets of partitions into $m$ equinumerous classes. The behavior for primes $m' \equiv 1 \pmod 6$ is also discussed., 28 pages, 19 figures

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.