Partition Congruence Research Articles

Congruences modulo powers of 5 for the crank parity function

In 1988, Andrews and Garvan introduced the partition statistic “crank” in order to give combinatorial interpretation for Ramanujan’s celebrated partition congruence modulo 11. In 2009, Choi, Kang and Lovejoy established congruences modulo powers of 5 for the crank parity function pC (n) by utilizing the theory of modular forms, where pC (n) denotes the difference between the number of partitions of n with even crank and the number of partitions of n with odd crank. In this paper, we provide a completely elementary proof of this congruence family. Moreover, we also prove several new congruences modulo small powers of 5.

A new approach to the Dyson rank conjectures

In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of $$5n+4$$ into five classes of equal size. This gave a combinatorial explanation of Ramanujan’s famous partition congruence mod 5. He made an analogous conjecture for the rank mod 7 and the partitions of $$7n+5$$ . In 1954 Atkin and Swinnerton-Dyer proved Dyson’s rank conjectures by constructing several Lambert-series identities basically using the theory of elliptic functions. In 2016 the author gave another proof using the theory of weak harmonic Maass forms. In this paper we describe a new and more elementary approach using Hecke–Rogers series.

Asymptotic distribution of the partition crank

Freeman Dyson conjectured the existence of an unknown partition statistic he called the crank which would explain Ramanujan’s partition congruence mod 11 just as his rank statistic explains Ramanujan’s partition congruences mod 5 and 7. Such a crank statistic was found by Andrews and Garvan in 1988. In this paper, we investigate the crank counting function, which counts the number of partitions of n with crank congruent to r mod Q. First, we obtain an effective bound on the error term in Zapata Rolón’s asymptotic formula for the crank counting function. We then use this to prove that the crank counting function is asymptotically equidistributed mod Q, for any odd number Q. We also use this to study surjectivity of the crank when viewed as a function from partitions to the integers mod Q, and to prove strict log-subadditivity of the crank counting function. The latter result is analogous to Bessenrodt and Ono’s strict log-subadditivity of the partition function.

Arithmetic properties of septic partition functions

Congruences and related identities are derived for a set of colored and weighted partition functions whose generating functions generate the graded algebra of integer weight modular forms of level seven. The work determines a general strategy for identifying and proving identities and associated congruences for modular forms on the principal congruence subgroup of level [Formula: see text]. Ramanujan’s partition congruence modulo [Formula: see text] serves as a prototype for the process used to prove new congruences for modular forms of level [Formula: see text].

Arithmetic properties of triangular partitions

We obtain a two variable generating function for the number of triangular partitions. Using this generating function, we study arithmetic properties of a certain weighted count of triangular partitions. Finally, we introduce a rank-type function for triangular partitions, which gives a combinatorial explanation for a triangular partition congruence.

A variation of the Andrews–Stanley partition function and two interesting q-series identities

Stanley introduced a partition statistic $$srank (\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$$ , where $$\mathcal {O}(\pi )$$ denote the number of odd parts of the partition $$\pi $$ , and $$\pi '$$ is the conjugate of $$\pi $$ . Let $$p_i(n)$$ denote the number of partitions of n with srank $$\equiv i\pmod 4$$ . Andrews proved the following refinement of Ramanujan’s partition congruence modulo 5: $$\begin{aligned} p_0(5n+4)\equiv p_2(5n+4)\equiv 0\pmod 5. \end{aligned}$$ In this paper, we consider an analogous partition statistic $$\begin{aligned} lrank (\pi )=\mathcal {O}(\pi )+\mathcal {O}(\pi '). \end{aligned}$$ Let $$p_i^+(n)$$ denote the number of partitions of n with lrank $$\equiv i \pmod 4$$ . We will establish the generating functions of $$p_0^+(n)$$ and $$p_2^+(n)$$ and show that they satisfy similar properties to $$p_i(n)$$ . We also utilize a pair of interesting q-series identities to obtain a direct proof of the congruences $$\begin{aligned} p_0^+(5n+4)\equiv p_2^+(5n+4)\equiv 0\pmod 5. \end{aligned}$$

Evaluating the role of morphological characters in the phylogeny of some trypanosomatid genera (Excavata, Kinetoplastea, Trypanosomatida).

Over the last three decades, it has been progressively assumed that morphology has become obsolete for trypanosomatid systematics. Traditional taxonomy, based on the occurrence of specific kinds of cell morphotypes during life cycles and the morphometry of such cells, is often rejected by molecular phylogenies inferred mostly from 18S rDNA alone or combined with GAPDH. In such context, we hypothesized the affinities of 35 representatives of seven trypanosomatid genera from separated and combined cladistics analyses of morphological and 18S matrices. Morphology is shown to be more consistent and to have stronger synapomorphy retention than the 18S data. The strict consensus of cladograms from separated analyses was mostly unresolved, while combined analysis produced a meaningful and robust phylogenetic pattern, as evidenced by partition congruence index, Bremer support and frequencies of groups present/contradicted. The results (1) corroborate the separation of Angomonas and Strigomonas from Crithidia, which is now shown to be monophyletic, (2) support the revalidation of the genus Wallaceina, and (3) place the genera Herpetomonas, Leptomonas and Phytomonas within a single clade. Overall, we demonstrate the belief that morphological characters are inferior to molecular ones for trypanosomatid systematics is a consequence of neglecting their inclusion in phylogenetic analyses.

Phylogenetic Utility of Mammalian Postcranial Characters

In phylogenetic studies of extinct species, researchers often exclude postcranial data because postcranial elements are missing, underrepresented, or difficult to assign to particular species. However, in some cases these decisions revolve around the idea that postcranial data are inappropriate for phylogenetic analyses because they are functionally or developmentally integrated or otherwise “prone to homoplasy.” In this study, we used a character congruence approach to test the hypothesis that postcranial data cannot be used to reconstruct phylogenies instead of the taxonomic congruence approaches most often employed in paleoanthropological studies. We identified 98 published mammalian phylogenetic datasets that included sufficient numbers of characters from the skull, dentition, and postcranium to be included in the analysis. We calculated partitioned Bremer support (PBS) values for each data partition on each strict consensus cladogram using the programs Phylogenetic Analysis Using Parsimony (PAUP) v. 4.0b10 and TreeRot.v3, calculated partition congruence indices (PCI), and assessed significant differences between data partitions using Fischer's least significant difference test. Averaged across each cladogram, postcranial PBS values are greater than, or not significantly different from, 69.4% of skull PBS values (68 studies) and 64.3% of dental PBS values (63 studies). Postcranial data provided primary support (“winning sites”) at 301/1448 nodes (20.8%), exhibited positive values at 710/1483 nodes (47.9%), and “do no harm” at 1196/1483 nodes (80.6%). There is no correlation between PBS values and data partition size as long as there are at least ten characters in each data partition. These results highlight the potential for using postcranial data to construct hominin phylogenies and argue against their exclusion from phylogenetic analyses for a priori reasons.Support or Funding InformationUndergraduate student research in this study was supported by the Benedictine University Natural Science Summer Research Program.

A short and simple proof of Ramanujan's mod 11 partition congruence

We present a short and simple proof of Ramanujan's partition congruence p(11n+6)≡0(mod11), and dedicate it to him on the occasion of his 125th birthday.

Phylogeny of the Liliales (Monocotyledons) with special emphasis on data partition congruence and RNA editing.

A phylogenetic analysis of the monocot order Liliales was performed using sequence data from three mitochondrial (atp1, cob, nad5) and two plastid genes (rbcL, ndhF). The complete data matrix includes 46 terminals representing all 10 families currently included in Liliales. The two major partitions, mitochondrial and plastid data, were congruent, and parsimony analysis resulted in 50 equally parsimonious trees and a well resolved consensus tree confirming monophyly of all families. Mitochondrial genes are known to include RNA edited sites, and in some cases unprocessed genes are replaced by retro-processed gene copies, that is processed paralogs. To test the effects on phylogeny reconstruction of predicted edited sites and potentially unintentionally sampled processed paralogs, a number of analyses were performed using subsets of the complete data matrix. In general, predicted edited sites were more homoplasious than the other characters and increased incongruence among most data partitions. The predicted edited sites have a non-random phylogenetic signal in conflict with the signal of the non-edited sites. The potentially misleading signal was caused partially by the apparent presence of processed paralogs in Galanthus (Amaryllidaceae), part of the outgroup, but also by a deviating evolutionary pattern of predicted edited sites in Liliaceae compared with the remainder of the Liliales. Despite the problems that processed paralogs may cause, we argue that they should not a priori be excluded from phylogenetic analysis.

Phylogenetic relationships among genera of danaine butterflies (Lepidoptera: Nymphalidae) as implied by morphology and DNA sequences

Cladistic relationships among genera and subtribes of Danaini (the milkweed butterflies) were inferred by analysis of data combined from five sources: morphology of adults and immature stages, and DNA sequences from three gene regions. The results corroborate and greatly increase support for prior hypotheses based on morphology alone. A new index summarizing incongruence among data partitions, the Partition Congruence Proportion (PCP), is introduced. The significance of the inferred pattern of phylogenetic relationships for comparative chemical ecology of milkweed butterflies is briefly discussed.

New identities for 7-cores with prescribed BG-rank

Let π be a partition. BG-rank ( π ) is defined as an alternating sum of parities of parts of π [A. Berkovich, F.G. Garvan, On the Andrews-Stanley refinement of Ramanujan's partition congruence modulo 5 and generalizations, Trans. Amer. Math. Soc. 358 (2006) 703–726. [1]]. Berkovich and Garvan [The BG-rank of a partition and its applications, Adv. in Appl. Math., to appear in 〈 http://arxiv.org/abs/math/0602362 〉 ] found theta series representations for the t-core generating functions ∑ n ⩾ 0 a t , j ( n ) q n , where a t , j ( n ) denotes the number of t-cores of n with BG-rank = j . In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme values of BG-rank. In this paper we discuss representations of this type for all 7-cores with prescribed BG-rank. We make an essential use of the Ramanujan modular equations of degree seven [B.C. Berndt, Ramanujan's Notebooks, Part III, Springer, New York, 1991] to prove a variety of new formulas for the 7-core generating function ∏ j ⩾ 1 ( 1 - q 7 j ) 7 ( 1 - q j ) . These formulas enable us to establish a number of striking inequalities for a 7 , j ( n ) with j = - 1 , 0 , 1 , 2 and a 7 ( n ) , such as a 7 ( 2 n + 2 ) ⩾ 2 a 7 ( n ) , a 7 ( 4 n + 6 ) ⩾ 10 a 7 ( n ) . Here a 7 ( n ) denotes a number of unrestricted 7-cores of n. Our techniques are elementary and require creative imagination only.

The BG-rank of a partition and its applications

Let π denote a partition into parts λ 1 ⩾ λ 2 ⩾ λ 3 ⩾ ⋯ . In a 2006 paper we defined BG-rank ( π ) as BG-rank ( π ) = ∑ j ⩾ 1 ( − 1 ) j + 1 1 − ( − 1 ) λ j 2 . This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let p j ( n ) denote the number of partitions of n with BG-rank = j . Here, we provide a combinatorial proof that p j ( 5 n + 4 ) ≡ 0 ( mod 5 ) , j ∈ Z , by showing that the residue of the 5-core crank mod 5 divides the partitions enumerated by p j ( 5 n + 4 ) into five equal classes. This proof uses the orbit construction from our previous paper and a new identity for the BG-rank. Let a t , j ( n ) denote the number of t-cores of n with BG-rank = j . We find eta-quotient representations for ∑ n ⩾ 0 a t , ⌊ t + 1 4 ⌋ ( n ) q n and ∑ n ⩾ 0 a t , − ⌊ t − 1 4 ⌋ ( n ) q n , when t is an odd, positive integer. Finally, we derive explicit formulas for the coefficients a 5 , j ( n ) , j = 0 , ± 1 .

Quintuple products and Ramanujan's partition congruence p(11n+6)≡0 (mod 11)

Quintuple products and Ramanujan's partition congruence p(11n+6)≡0 (mod 11)

The how and why of branch support and partitioned branch support, with a new index to assess partition incongruence.

An introduction and step-by-step guide to the calculation of branch support (BS) and partitioned branch support (PBS) is provided, along with a new summary measure of conflict among data partitions, the partition congruence index (PCI). Patterns of PCI vs. BS are compared for two data matrices from butterflies. PCI is a close mirror of BS unless BY is relatively low and there is a lot of conflict among data partitions, in which case PCI becomes negative. Some theoretical concerns relating to PBS are noted.

Phylogenetic relationships among the Ithomiini (Lepidoptera: Nymphalidae) inferred from one mitochondrial and two nuclear gene regions

Abstract. A phylogenetic hypothesis for the tribe Ithomiini (Lepidoptera: Nymphalidae: Danainae) is presented, based on sequences of the mitochondrial cytochrome oxidase subunits I and II (COI–COII) region and regions of the nuclear genes wingless and Elongation factor 1‐alpha. Branch support for each clade is assessed, and a partition congruence index is used to explore conflict among gene regions. The monophyly of the clade is strongly supported, as are many of the traditionally recognized subtribes and genera. The data imply paraphyly of some genera and tribes, but largely support recent classifications and phylogenetic hypotheses based on morphological characters.

On the Andrews-Stanley refinement of Ramanujan’s partition congruence modulo 5 and generalizations

In a recent study of sign-balanced, labelled posets, Stanley introduced a new integral partition statistic s r a n k ( π ) = O ( π ) − O ( π ′ ) , \begin{equation*} \mathrm {srank}(\pi ) = {\mathcal O}(\pi ) - {\mathcal O}(\pi ’), \end{equation*} where O ( π ) {\mathcal O}(\pi ) denotes the number of odd parts of the partition π \pi and π ′ \pi ’ is the conjugate of π \pi . In a forthcoming paper, Andrews proved the following refinement of Ramanujan’s partition congruence mod 5 5 : p 0 ( 5 n + 4 ) a m p ; ≡ p 2 ( 5 n + 4 ) ≡ 0 ( mod 5 ) , p ( n ) a m p ; = p 0 ( n ) + p 2 ( n ) , \begin{align*} p_0(5n+4) &\equiv p_2(5n+4) \equiv 0 \pmod {5}, p(n) &= p_0(n) + p_2(n), \end{align*} where p i ( n ) p_i(n) ( i = 0 , 2 i=0,2 ) denotes the number of partitions of n n with s r a n k ≡ i ( mod 4 ) \mathrm {srank}\equiv i\pmod {4} and p ( n ) p(n) is the number of unrestricted partitions of n n . Andrews asked for a partition statistic that would divide the partitions enumerated by p i ( 5 n + 4 ) p_i(5n+4) ( i = 0 , 2 i=0,2 ) into five equinumerous classes. In this paper we discuss three such statistics: the ST-crank, the 2 2 -quotient-rank and the 5 5 -core-crank. The first one, while new, is intimately related to the Andrews-Garvan (1988) crank. The second one is in terms of the 2 2 -quotient of a partition. The third one was introduced by Garvan, Kim and Stanton in 1990. We use it in our combinatorial proof of the Andrews refinement. Remarkably, the Andrews result is a simple consequence of a stronger refinement of Ramanujan’s congruence mod 5 5 . This more general refinement uses a new partition statistic which we term the BG-rank. We employ the BG-rank to prove new partition congruences modulo 5 5 . Finally, we discuss some new formulas for partitions that are 5 5 -cores and discuss an intriguing relation between 3 3 -cores and the Andrews-Garvan crank.

A New Identity for (q;q) 10 with an Application to Ramanujan's Partition Congruence Modulo 11

Our principal second goal is to show that this identity leads to a short proof of Ramanujan’s famous congruence p(11n+6) ≡ 0 (mod 11), where p(n) denotes the number of unrestricted partitions of the positive integer n. Our proof of Theorem 2.2 is short but depends upon some results of Ramanujan from his notebooks [18]. In Section 4, we give a more elementary proof, based upon Ramanujan’s 1ψ1 summation formula, of Ramanujan’s principal result, Lemma 2.1, which is employed in our proof of Theorem 2.2. In Section 5, we give an entirely different and direct proof of Theorem 2.2 based on several elementary identities for theta functions due to Ramanujan. In fact, during our first proof of Theorem 2.2, we establish a special case of a general result on Eisenstein series found in Ramanujan’s lost notebook [19, p. 369]. More precisely, Ramanujan asserts that every member of a certain class of infinite series can be expressed in terms of Ramanujan’s Eisenstein series P , Q, and R (to be defined in Section 6). A less precise version of this claim appears in Ramanujan’s notebooks [18], [2, p. 65, Entry 35(i)], but we prove the better version in Section 6. D. Stanton empirically discovered an analogue of Theorem 2.2, and in Section 7 we give a proof of Stanton’s identity. In Section 8, we prove that (q; q) ∞ is lacunary; in this connection, see a result of J.–P. Serre [20].

A New Identity for (q;q) 10 with an Application to Ramanujan's Partition Congruence Modulo 11

A New Identity for (q;q) 10 with an Application to Ramanujan's Partition Congruence Modulo 11

More cranks andt-cores

Dedicated to George Szekeres on the occasion of his 90th BirthdayIn 1990, new statistics on partitions (calledcransk) were found which combinatorially prove Ramanujan's congruences for the partition function modulo 5, 7, 11 and 25. The methods are extended to find cranks for Ramanujan's partition congruence modulo 49. A more explicit form of the crank is given for the modulo 25 congruence.