Unrestricted Partitions Research Articles

Combinatorial Constructions of Generating Functions of Cylindric Partitions with Small Profiles into Unrestricted or Distinct Parts

In this paper, cylindric partitions into profiles $c=(1,1)$ and $c=(2,0)$ are considered. The generating functions into unrestricted cylindric partitions and cylindric partitions into distinct parts with these profiles are constructed. The constructions are combinatorial and they connect the cylindric partitions with ordinary partitions.

Some identities for the partition function

In his unpublished manuscript on the partition and tau functions, Ramanujan obtained several striking congruences for the partition function p(n), the number of unrestricted partitions of n. The most notable of them are p(5n+4)≡0(mod5) and p(7n+5)≡0(mod7) which holds for all positive integers n. More surprisingly, Ramanujan obtained certain identities between q-series from which the above congruences follow as consequences. In this paper, we adopt Ramanujan's approach and prove an identity which witnesses another famous Ramanujan congruence, namely, p(11n+6)≡0(mod11) and also establish some new identities for the generating functions for p(17n+5),p(19n+7) and p(23n+1). We also find explicit evaluations for Fp(q) in the cases p=17,19,23 where Fp is the function appearing in Ramanujan's circular summation formula.

Congruences for $\ell$-regular partitions and bipartitions

Define F ( q ) : = ∑ n = − ∞ ∞ ( − 1 ) δ n ( a n + b ) q ( c n 2 + d n ) ∕ 2 , which includes Ramanujan’s theta function as a special case. We establish a dissection identity for this function, and use it to derive congruence properties for the coefficients of F ( q ) . As an application we deduce several infinite families of congruences for ℓ -regular partitions and ℓ -regular bipartitions. In addition, we give a new proof of Ramanujan’s congruence for the unrestricted partition function modulo 5 .

On a new formula for the number of unrestricted partitions

In this paper we present a new formula for the number of unrestricted partitions of n. We do this by introducing a correspondence between the number of unrestrited partitions of n and the number of non-negative solutions of systems of two equations, involving natural numbers in the interval $$(1 ,n^{2})$$ .

On 5-regular bipartitions with even parts distinct

In 2010, Andrews, Michael D. Hirschhorn and James A. Sellers considered the function ped(n), the number of partition of an integer n with even parts distinct (the odd parts are unrestricted). They obtained infinite families of congruences in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). Let b(n) denote the number of 5-regular bipartitions of a positive integer n with even parts distinct (odd parts are unrestricted). In this paper, we establish many infinite families of congruences modulo powers of 2 for b(n). For example, $$\begin{aligned} \sum _{n=0}^{\infty } b\left(16\cdot 3^{2\alpha }\cdot 5^{2\beta }n+14\cdot 3^{2\alpha }\cdot 5^{2\beta }+1\right) q^n \equiv 8f_2^3f_5^3 \pmod {16} , \end{aligned}$$where $$\alpha , \beta \ge 0$$.

Partitions into kth powers of terms in an arithmetic progression

G. H. Hardy and S. Ramanujan established an asymptotic formula for the number of unrestricted partitions of a positive integer, and claimed a similar asymptotic formula for the number of partitions into perfect kth powers, which was later proved by E. M. Wright. Recently, R. C. Vaughan provided a simpler asymptotic formula in the case $$k=2$$ . In this paper, we consider partitions into parts from a specific set $$A_k(a_0,b_0) :=\left\{ m^k : m \in \mathbb {N} , m \equiv a_0 \,(\text {mod}\,b_0) \right\} $$ , for fixed positive integers k, $$a_0,$$ and $$b_0$$ . We give an asymptotic formula for the number of such partitions, thus generalizing the results of Wright and Vaughan. Moreover, we prove that the number of such partitions is even (odd) infinitely often, which generalizes O. Kolberg’s theorem for the ordinary partition function p(n).

A general asymptotic formula for distinct partitions

Many asymptotic formulas exist for unrestricted integer partitions as well as for equal partitions of integers into a finite number of parts. We use an analogy with fermion gases and the tools of statistical physics to derive asymptotic formulas for distinct partitions with a large but finite number of parts. These results are supported by the fact that we recover some other existing asymptotic results and by numerical comparisons with exact results.

A New Approach to Integer Partitions

In this work we define a new set of integer partition, based on a lattice path in $${\mathbb {Z}}^2$$ connecting the line $$x+y=n$$ to the origin, which is determined by the two-line matrix representation given for different sets of partitions of n. The new partitions have only distinct odd parts with some particular restrictions. This process of getting new partitions, which has been called the Path Procedure, is applied to unrestricted partitions, partitions counted by the 1st and 2nd Rogers–Ramanujan Identities, and those generated by the Mock Theta Function $$T_1^*(q)=\sum _{n=0}^{\infty }\dfrac{q^{n(n+1)}(-q^2,q^2)_n}{(q,q^2)_{n+1}}$$ .

The Riemann Zeta Function With Even Arguments as Sums Over Integer Partitions

In this note, we build on recent work in [7] to establish formulas for ζ(2n) as sums over all the unrestricted integer partitions of n.

Asymptotic prime partitions of integers.

In this paper, we discuss P(n), the number of ways a given integer n may be written as a sum of primes. In particular, an asymptotic form P_{as}(n) valid for n→∞ is obtained analytically using standard techniques of quantum statistical mechanics. First, the bosonic partition function of primes, or the generating function of unrestricted prime partitions in number theory, is constructed. Next, the density of states is obtained using the saddle-point method for Laplace inversion of the partition function in the limit of large n. This gives directly the asymptotic number of prime partitions P_{as}(n). The leading term in the asymptotic expression grows exponentially as sqrt[n/ln(n)] and agrees with previous estimates. We calculate the next-to-leading-order term in the exponent, proportional to ln[ln(n)]/ln(n), and we show that an earlier result in the literature for its coefficient is incorrect. Furthermore, we also calculate the next higher-order correction, proportional to 1/ln(n) and given in Eq.(43), which so far has not been available in the literature. Finally, we compare our analytical results with the exact numerical values of P(n) up to n∼8×10^{6}. For the highest values, the remaining error between the exact P(n) and our P_{as}(n) is only about half of that obtained with the leading-order approximation. But we also show that, unlike for other types of partitions, the asymptotic limit for the prime partitions is still quite far from being reached even for n∼10^{7}.

A General Asymptotic Formula for Distinct Partitions

Many asymptotic formulas exist for unrestricted integer partitions as well as for equal partitions of integers into a finite number of parts. We use an analogy with fermion gases and the tools of statistical physics to derive asymptotic formulas for distinct partitions with a large but finite number of parts. These results are supported by the fact that we recover some other existing asymptotic results and by numerical comparisons with exact results.

Asymptotics and congruences for partition functions which arise from finitary permutation groups

In a recent paper, Bacher and de la Harpe study the conjugacy growth series of finitary permutation groups. In the course of studying the coefficients of a series related to the finitary alternating group, they introduce generalized partition functions $$p(n)_\mathbf{e }$$ . The group theory motivates the study of the asymptotics for these functions. Moreover, Bacher and de la Harpe also conjecture over 200 congruences for these functions which are analogous to the Ramanujan congruences for the unrestricted partition function p(n). We obtain asymptotic formulas for all of the $$p(n)_\mathbf{e }$$ and prove their conjectured congruences.

Some identities for derangement and Ward number sequences and related bijections

Abstract We establish an alternating sum identity for three classes of singleton-free set partitions wherein the number of elements minus the number of blocks is fixed: (i) permutations, that is, partitions into cycles, (ii) unrestricted partitions, and (iii) contents-ordered partitions. Both algebraic and combinatorial proofs are given, the latter making use of a sign-changing involution in each ease. As a consequence, combinatorial proofs are found of specific cases of recent identities of Gould et al. involving both kinds of Stirling numbers.

An infinite family of congruences modulo $$3$$ 3 for $$13$$ 13 -regular bipartitions

Let \(B_{13}(n)\) denote the number of \(13\)-regular bipartitions of \(n\). Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function \(p(n)\). In particular, we shall prove an infinite family of congruences: for \(\alpha \ge 2\) and \(n\ge 0\), $$\begin{aligned} B_{13}(3^{\alpha }n+2\cdot 3^{\alpha -1}-1)\equiv \ 0\ (\mathrm{mod\ }3). \end{aligned}$$ In addition, we will also give an alternative proof of one infinite family of congruences for \(b_{13}(n)\), the number of \(13\) regular partitions of \(n\), due to Webb.

Arithmetic of the 7-regular bipartition function modulo 3

In this paper, we study the function B l (n) which counts the number of l-regular bipartitions of n. Our goal is to consider this function from an arithmetical point of view in the spirit of Ramanujan’s congruences for the unrestricted partition function p(n). In particular, using Ramanujan’s two modular equations of degree 7, we prove an infinite family of congruences: for α≥2 and n≥0, $$B_7 \biggl(3^{\alpha}n+\frac{5\cdot 3^{\alpha-1}-1}{2} \biggr)\equiv 0\ ({ \rm mod\ }3). $$ In addition, we give an elementary proof of two infinite families of congruences modulo 3 satisfied by the 7-regular partition function due to Furcy and Penniston (Ramanujan J. 27:101–108, 2012). We also present two conjectures for B 13(n) modulo 3.

On the Distribution of the spt-Crank

Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence { N S (m , n) } m is unimodal, where N S (m , n) is the number of S-partitions of size n with crank m weight by the spt-crank. We relate this conjecture to a distributional result concerning the usual rank and crank of unrestricted partitions. This leads to a heuristic that suggests the conjecture is true and allows us to asymptotically establish the conjecture. Additionally, we give an asymptotic study for the distribution of the spt-crank statistic. Finally, we give some speculations about a definition for the spt-crank in terms of “marked” partitions. A “marked” partition is an unrestricted integer partition where each part is marked with a multiplicity number. It remains an interesting and apparently challenging problem to interpret the spt-crank in terms of ordinary integer partitions.

The crank of partitions mod 13

Let M(r,m,n) be the number of unrestricted partitions of n with crank congruent to r modulo m. Here the relations between the numbers M(r,13,13n+k) when 0≤r<13 are given. All of the results are proved by elementary methods.

Generalized congruence properties of the restricted partition function p(n,m)

Ramanujan-type congruences for the unrestricted partition function p(n) are well known and have been studied in great detail. The existence of Ramanujan-type congruences are virtually unknown for p(n,m), the closely related restricted partition function that enumerates the number of partitions of n into exactly m parts. Let l be any odd prime. In this paper we establish explicit Ramanujan-type congruences for p(n,l) modulo any power of that prime lα. In addition, we establish general congruence relations for p(n,l) modulo lα for any n.

Local distribution of the parts of unequal partitions in arithmetic progressions I

In [4], Andras Sarkozy and the authors proved that for almost all unequal partitions of an integer n, the parts are evenly distributed in residue classes modulo d for d = o(n^(1/2)). In this paper, we study very precisely the local distribution in arithmetic progressions of the parts of unequal partitions. We obtain some asymptotic formulae for the number of unequal partitions of n with exactly N r parts congruent to r mod d, 1 6 r 6 d. Our results show that (N 1 ,. .. , N d) behaves like a random Gaussian vector. This illustrates the fact that the distribution of the parts of unequal partitions in residue classes is much more uniform that in the case of unrestricted partitions.

A SHORT PROOF OF AN IDENTITY FOR CUBIC PARTITION FUNCTION

Abstract. In this note, we will give a short proof of an identity for cubicpartition function. 1. Introduction∑Let p ( n ) denote the number of the unrestricted partitions of n , de ned by 1n =0 p ( n ) q n =∏ 1n =011 q n . One of the celebrated results about p ( n ) is thetheorem which was proved by Watson [8]: if k  1, then for every nonnegativeinteger n (1) p (5 k n + r k )  0 (mod 5 k ) ; where r k is the reciprocal modulo 5 k of 24. Recently, the notion of cubicpartitions of a natural number n , named by Kim [5], was introduced by Chan[1] in connection with Ramanujan’s cubic continued fraction. By de ntion, thegenerating function of the number of cubic partitions of n is(2)∑ 1n =0 a ( n ) q n =∏ 1n =1 1(1 q n )(1 q 2 n ) : Chan [1] from the Ramanujan’s cubic continued fraction v ( q ) := q 13 1 + q + q 3 1 + q 2 + q 4 1 + ::: jqj  1derived an elegant identity: let x ( q ) = q 13 v ( q ), then1 x ( q ) q 13 2 q 23 x ( q ) =( q 13 ; q 13 ) 1 ( q 23 ; q 23 ) 1 ( q 3 ; q 3 ) 1 ( q