Abstract. We present an extensive survey of bijective proofs of classical partitions identities. While most bijections are known, they are often presented in a different, sometimes unrecognizable way. Various extensions and generalizations are added in the form of exercises.

As usual, define Dedekind’s eta-function η(z) by the infinite product η(z): = q 1/24 n 1 − q) ( q: = e 2πiz throughout). n=1 In a recent paper, D. Zagier proved that (note: empty products equal 1 throughout) n=0 η(24z) − q ( 1 − q 24) ( 1 − q 48) ·· · ( 1 − q 24n)) = η(24z)D(q) + E(q), where the series D(q) and E(q) are defined by D(q) = − 1 2 + E(q) = 1

Let F(n) be a family of partitions of n and let F(n; d) denote the set of partitions in F(n) with Durfee square of size d. We define the Durfee polynomial of F(n) to be the polynomial PF;n = P jF(n; d)jy d , where 0 d b p nc: The work in this paper is motivated by empirical evidence which suggests that for several families F, all roots of the Durfee polynomial are real. Such a result would imply that the corresponding sequence of coefficients fjF(n; d)jg is logconcave and unimodal and that, over all partitions in F(n) for fixed n, the average size of the Durfee square, aF (n), and the most likely size of the Durfee square, mF (n), differ by less than 1. In this paper, we prove results in support of the conjecture that for the family of ordinary partitions, P(n), the Durfee polynomial has all roots real. Specifically, we find an asymptotic formula for jP(n; d)j, deriving in the process a simple upper bound on the number of partitions of n with at most k parts which generalizes the upper bound of Erdos for jP(n)j. We show that as n tends to infinity, the sequence fjP(n; d)jg; 1 d p n; is asymptotically normal, unimodal, and log concave; in addition, formulas are found for aP (n) and mP (n) which differ asymptotically by at most 1. Experimental evidence also suggests that for several families F(n) which satisfy a recurrence of a certain form, mF (n) grows as c p n, for an appropriate constant c = c F . We prove this under an assumption about the asymptotic form of jF(n; d)j and show how to produce, from recurrences for the jF(n; d)j, analyti...

Let P+,(n) denote the number of partitions of n into summands chosen from the set A = (a,. n2, I..}. De Bruijn has shown that in Mahler’s partition problem (a ” = I”) there is a periodic component in the asymptotic behaviour of P,,(n). We show by example that this may happen For sequences that satisfy a,- v and consider an analogous phenomena for partitions into primes. We then consider corresponding results for partitions into distinct summands. Finally we obtain some weaker results using elementary methods. 1.

Let m ≥ 1, and let A be the set of all positive integers that belong to a union of ℓ distinct congruence classes modulo m. Let pA(n) denote the number of partitions of n into parts belonging to A. It is proved that

In this paper we specialize work done by Bateman and Erdős concerning difference functions of partition functions. In particular we are concerned with partitions into fixed powers of the primes. We show that any difference function of these partition functions is eventually increasing, and derive explicit bounds for when it will attain strictly positive values. From these bounds an asymptotic result is derived.

Abstract. In 2003, Maróti showed that one could use the machinery of ℓ-cores and ℓ-quotients of partitions to establish lower bounds for p(n), the number of partitions of n. In this paper we explore these ideas in the case ℓ = 2, using them to give a largely combinatorial proof of an effective upper bound on p(n), and to prove asymptotic formulae for the number of self-conjugate partitions, and the number of partitions with distinct parts. In a further application we give a combinatorial proof of an identity originally due to Gauss. 1.

Let A be a set of positive integers with gcd(A) = 1, and let pA(n) be the partition function of A. Let c0 = π √ 2/3. If A has lower asymptotic density α and upper asymptotic density β, then lim inf log pAn/c0 n ≥ α and lim sup log pA(n)/c0 n ≤ β. In particular, if A has asymptotic density α> 0, then log pA(n) ∼ c0 αn. Conversely, if α> 0 and log pA(n) ∼ c0 αn, then the set A has asymptotic density α. 1 The growth of pA(n) Let A be a nonempty set of positive integers. The counting function A(x) of the set A counts the number of positive elements of A that do not exceed x. Then 0 ≤ A(x) ≤ x, and so 0 ≤ A(x)/x ≤ 1 for all x. The lower asymptotic density of A is dL(A) = liminf x→∞ The upper asymptotic density of A is dU(A) = limsup x→∞ A(x) x.