Some congruences for Andrews-Paule’s broken 2-diamond partitions

We prove two conjectures of Andrews and Paule [G.E. Andrews, P. Paule, MacMahon’s partition analysis XI: Broken diamonds and modular forms, Acta Arith. 126 (2007) 281-294] on congruences of broken k-diamond partitions.     

Congruences for Broken k-Diamond Partitions

We prove two conjectures of Paule and Radu from their recent paper on broken k-diamond partitions.     

Congruences modulo 4 for broken <Emphasis Type="Italic">k</Emphasis>-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, a number of parity results satisfied by \(\Delta _k(n)\) for small values of k have been proved by Radu and Sellers and others. However, congruences modulo 4 for \(\Delta _k(n)\) are unknown. In this paper, we will prove five congruences modulo 4 for \(\Delta _5(n)\), four infinite families of congruences modulo 4 for \(\Delta _7(n)\) and one congruence modulo 4 for \(\Delta _{11}(n)\) by employing theta function identities. Furthermore, we will prove a new parity result for \(\Delta _2(n)\).     

Infinite families of strange partition congruences for broken 2-diamonds

In 2007, George E. Andrews and Peter Paule (Acta Arithmetica 126:281–294, 2007) introduced a new class of combinatorial objects called broken k-diamonds. Their generating functions connect to modular forms and give rise to a variety of partition congruences. In 2008, Song Heng Chan proved the first infinite family of congruences when k=2. In this note, we present two non-standard infinite families of broken 2-diamond congruences derived from work of Oliver Atkin and Morris Newman. In addition, four conjectures related to k=3 and k=5 are stated.     

On the parity of broken k-diamond partitions

In this paper, we prove several new parity results for broken \(k\) -diamond partitions on certain types of arithmetic progressions. We also obtain bounds for the parity of broken \(k\) -diamond partitions and more general colored partitions.     

Parity results for 13-regular partitions and broken 6-diamond partitions

Let \(b_{13}(n)\) denote the number of partitions of n such that no parts are divisible by 13. In this paper, we shall prove several infinite families of congruence relations modulo 2 for \(b_{13}(n)\). In addition, we will give an elementary proof of the parity result on broken 6-diamond partitions, which was established by Radu and Sellers. We also find a new congruence relation modulo 2 for the number of broken 6-diamond partitions of n.     

Infinite families of congruences modulo 7 for broken 3-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu conjectured that \(\Delta _3(343n+82)\equiv \Delta _3(343n+278)\equiv \Delta _3(343n+327)\equiv 0\ (\mathrm{mod} \ 7)\). Jameson confirmed this conjecture and proved that \(\Delta _3(343n+229)\equiv 0 \ (\mathrm{mod} \ 7)\) by using the theory of modular forms. In this paper, we prove several infinite families of Ramanujan-type congruences modulo 7 for \(\Delta _3(n)\) by establishing a recurrence relation for a sequence related to \(\Delta _3(7n+5)\). In the process, we also give new proofs of the four congruences due to Paule and Radu, and Jameson.     

Ramanujan-type congruences for broken 2-diamond partitions modulo 3

The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).     

Free Skew Boolean Intersection Algebras and Set Partitions

We show that atoms of the n-generated free left-handed skew Boolean intersection algebra are in a bijective correspondence with pointed partitions of non-empty subsets of \(\{1,2,\dots , n\}\). Furthermore, under the canonical inclusion into the k-generated free algebra, where k≥n, an atom of the n-generated free algebra decomposes into an orthogonal join of atoms of the k-generated free algebra in an agreement with the containment order on the respective partitions. As a consequence of these results, we describe the structure of finite free left-handed skew Boolean intersection algebras and express several their combinatorial characteristics in terms of Bell numbers and Stirling numbers of the second kind. We also look at the infinite case. For countably many generators, our constructions lead to the ‘partition analogue’ of the Cantor tree whose boundary is the ‘partition variant’ of the Cantor set.     

Newman’s identity and infinite families of congruences modulo 7 for broken 3-diamond partitions

In 2007, Andrews and Paule introduced the notion of broken k-diamond partitions. Let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n for a fixed positive integer k. Recently, Paule and Radu presented some conjectures on congruences modulo 7 for \(\Delta _3(n)\) which were proved by Jameson and Xiong based on the theory of modular forms. Very recently, Xia proved several infinite families of congruences modulo 7 for \(\Delta _3(n)\) using theta function identities. In this paper, many new infinite families of congruences modulo 7 for \(\Delta _3(n)\) are derived based on an identity of Newman and the (p; k)-parametrization of theta functions due to Alaca, Alaca and Williams. In particular, some non-standard congruences modulo 7 for \(\Delta _3(n)\) are deduced. For example, we prove that for \(\alpha \ge 0\), \(\Delta _3\left( \frac{14\times 757^{\alpha }+1}{3}\right) \equiv 6 -\alpha \ (\mathrm{mod}\ 7)\).     

Broken 2-diamond partitions modulo 5

We consider \(\Delta _2(n)\), the number of broken 2-diamond partitions of n, and give simple proofs of two congruences given by Song Heng Chan.     

Multiplicative Properties of the Number of <Emphasis Type="Italic">k</Emphasis>-Regular Partitions

In a previous paper of the second author with K. Ono, surprising multiplicative properties of the partition function were presented. Here, we deal with k-regular partitions. Extending the generating function for k-regular partitions multiplicatively to a function on k-regular partitions, we show that it takes its maximum at an explicitly described small set of partitions, and can thus easily be computed. The basis for this is an extension of a classical result of Lehmer, from which an inequality for the generating function for k-regular partitions is deduced which seems not to have been noticed before.     

Congruences modulo 11 for broken 5-diamond partitions

The notion of broken k-diamond partitions was introduced by Andrews and Paule in 2007. For a fixed positive integer k, let \(\Delta _k(n)\) denote the number of broken k-diamond partitions of n. Recently, Paule and Radu conjectured two relations on \(\Delta _5(n)\) which were proved by Xiong and Jameson, respectively. In this paper, employing these relations, we prove that, for any prime p with \(p\equiv 1\ (\mathrm{mod}\ 4)\), there exists an integer \(\lambda (p)\in \{2,\ 3,\ 5,\ 6,\ 11\}\) such that, for \(n, \alpha \ge 0\), if \(p\not \mid (2n+1)\), then

$$\begin{aligned} \Delta _5\left( 11p^{\lambda (p)(\alpha +1)-1} n+\frac{11p^{\lambda (p)(\alpha +1)-1}+1}{2}\right) \equiv 0\ (\mathrm{mod}\ 11). \end{aligned}$$

Moreover, some non-standard congruences modulo 11 for \(\Delta _5(n)\) are deduced. For example, we prove that, for \(\alpha \ge 0\), \(\Delta _5\left( \frac{11\times 5^{5\alpha }+1}{2}\right) \equiv 7\ (\mathrm{mod}\ 11)\).

     

Elementary proofs of infinitely many congruences for k-elongated partition diamonds

In 2007, Andrews and Paule published the eleventh paper in their series on MacMahon's partition analysis, with a particular focus on broken k-diamond partitions. On the way to broken k-diamond partitions, Andrews and Paule introduced the idea of k-elongated partition diamonds. Recently, Andrews and Paule revisited the topic of k-elongated partition diamonds. Using partition analysis and the Omega operator, they proved that the generating function for the partition numbers $d_k(n)$ produced by summing the links of k-elongated plane partition diamonds of length n is given by $\frac{{(q^2;q^2)}_{\infty }^k}{{(q;q)}_{\infty }^{3k+1}}$ for each $k\ge 1$. A significant portion of their recent paper involves proving several congruence properties satisfied by $d_1,d_2$ and $d_3$, using modular forms as their primary proof tool. In this work, our goal is to extend some of the results proven by Andrews and Paule in their recent paper by proving infinitely many congruence properties satisfied by the functions $d_k$ for an infinite set of values of k. The proof techniques employed are all elementary, relying on generating function manipulations and classical q-series results.     

Partitions with rounded occurrences and attached parts

We introduce the number of (k,i)-rounded occurrences of a part in a partition and use q-difference equations to interpret a certain q-series S _k,i (a;x;q) as the generating function for partitions with bounded (k,i)-rounded occurrences and attached parts. When a=0 these partitions are the same as those studied by Bressoud in his extension of the Rogers-Ramanujan-Gordon identities to even moduli. When a=1/q we obtain a new family of partition identities.     

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.     

Some inequalities for <Emphasis Type="Italic">k</Emphasis>-colored partition functions

Motivated by a partition inequality of Bessenrodt and Ono, we obtain analogous inequalities for k-colored partition functions \(p_{-k}(n)\) for all \(k\ge 2\). This enables us to extend the k-colored partition function multiplicatively to a function on k-colored partitions and characterize when it has a unique maximum. We conclude with one conjectural inequality that strengthens our results.     

Standard simplices and pluralities are not the most noise stable

The Standard Simplex Conjecture and the Plurality is Stablest Conjecture are two conjectures stating that certain partitions are optimal with respect to Gaussian and discrete noise stability respectively. These two conjectures are natural generalizations of the Gaussian noise stability result by Borell (1985) and the Majority is Stablest Theorem (2004). Here we show that the standard simplex is not the most stable partition in Gaussian space and that Plurality is not the most stable low influence partition in discrete space for every number of parts k ≥ 3, for every value ρ ≠ 0 of the noise and for every prescribed measure for the different parts as long as they are not all equal to 1/k. Our results do not contradict the original statements of the Plurality is Stablest and Standard Simplex Conjectures in their original statements concerning partitions to sets of equal measure. However, they indicate that if these conjectures are true, their veracity and their proofs will crucially rely on assuming that the sets are of equal measures, in stark contrast to Borell’s result, the Majority is Stablest Theorem and many other results in isoperimetric theory. Given our results it is natural to ask for (conjectured) partitions achieving the optimum noise stability.     

Parity of sums of partition numbers and squares in arithmetic progressions

In this paper, we prove new infinite families of congruences modulo 2 for broken 11-diamond partitions by using Hecke operators.     

Building initial partitions through sampling techniques

A variety of iterative clustering algorithms require an initial partition of a dataset as an input parameter. As a rule a good choice of the initial partition is essential for building a high quality final partition. In this note, we generate initial partitions by using small samples of the data. Numerical experiments with k-means like clustering algorithms are reported.