Touchard–Riordan formulas, T-fractions, and Jacobi’s triple product identity

Touchard–Riordan-like formulas are certain expressions appearing in enumeration problems and as moments of orthogonal polynomials. We begin this article with a new combinatorial approach to prove such formulas, related to integer partitions. This gives a new perspective on the original result of Touchard and Riordan. But the main goal is to give a combinatorial proof of a Touchard–Riordan-like formula for q-secant numbers discovered by the first author. An interesting limit case of these objects can be directly interpreted in terms of partitions, so that we obtain a connection between the formula for q-secant numbers, and a particular case of Jacobi’s triple product identity. Building on this particular case, we obtain a “finite version” of the triple product identity. It is in the form of a finite sum which is given a combinatorial meaning, so that the triple product identity can be obtained by taking the limit. Here the proof is non-combinatorial and relies on a functional equation satisfied by a T-fraction. Then from this result on the triple product identity, we derive a whole new family of Touchard–Riordan-like formulas whose combinatorics is not yet understood. Eventually, we prove a Touchard–Riordan-like formula for a q-analog of Genocchi numbers, which is related with Jacobi’s identity for (q;q)³ rather than the triple product identity.     

On a partition identity of Lehmer

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called “Beck-type” companions to other identities.In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.     

Combinatorial telescoping for an identity of Andrews on parity in partitions

Recently Andrews proposed a problem of finding a combinatorial proof of an identity on the q-little Jacobi polynomials. We give a classification of certain triples of partitions and find bijections based on this classification. By the method of combinatorial telescoping for identities on sums of positive terms, we establish a recurrence relation that leads to the identity of Andrews.     

Limiting shapes of birth-and-death processes on Young diagrams

We consider a family of birth processes and birth-and-death processes on Young diagrams of integer partitions of n. This family incorporates three famous models from very different fields: Rost?s totally asymmetric particle model (in discrete time), Simon?s urban growth model, and Moran?s infinite alleles model. We study stationary distributions and limit shapes as n tends to infinity, and present a number of results and conjectures.     

Refinements of the results on partitions and overpartitions with bounded part differences

Recently, partitions with fixed or bounded differences between largest and smallest parts have attracted a lot of attention. In this paper, we first give a simple combinatorial proof of Breuer and Kronholm’s identity. Inspired by it, we construct a useful bijection to produce refinements of the results for partitions and overpartitions with bounded differences between largest and smallest parts. Consequently, we obtain Chern’s curious identity in a combinatorial manner.     

The minimal excludant and Schmidt's partition theorem

In this work, we study Schmidt's partition theorem in a combinatorial manner, and find a strong refinement which connects the minimal excludant of ordinary partitions to the length of Schmidt's partitions. As a byproduct, we obtain a bivariate form of an identity recorded in Ramanujan's lost notebook.     

On flushed partitions and concave compositions

In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed partitions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews’ partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order f(q), ?(q) and ψ(q). An identity of Ramanujan is proved combinatorially. Several new identities are also established.     

Combinatorial interpretations of congruences for the spt-function

Let spt(n) denote the total number of appearances of the smallest parts in all the partitions of n. In 1988, the second author gave new combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11 in terms of a crank for weighted vector partitions. In 2008, the first author found Ramanujan-type congruences for the spt-function mod 5, 7 and 13. We give new combinatorial interpretations of the spt-congruences mod 5 and 7. These are in terms of the same crank but for a restricted set of vector partitions. The proof depends on relating the spt-crank with the crank of vector partitions and the Dyson rank of ordinary partitions. We derive a number of identities for spt-crank modulo 5 and 7. We prove the surprising result that all the spt-crank coefficients are nonnegative.     

Com plete p-elli ptic integrals and a com putation formula of $$\ pi _ p$$π p for $$ p=4$$ p=4

We study the q-bracket operator of Bloch and Okounkov, recently examined by Zagier and other authors, when applied to functions defined by two classes of sums over the parts of an integer partition. We derive convolution identities for these functions and link both classes of q-brackets through divisor sums. As a result, we generalize Euler’s classic convolution identity for the partition function and obtain an analogous identity for the totient function. As corollaries, we generalize Stanley’s Theorem on the number of ones in all partitions of n, and provide several new combinatorial results.     

Euler-Mahonian statistics on ordered set partitions (II)

[E. Steingrímsson, Statistics on ordered partitions of sets, arXiv: math.CO/0605670] introduced several hard statistics on ordered set partitions and conjectured that their generating functions are related to the q-Stirling numbers of the second kind. In a previous paper, half of these conjectures have been proved by Ishikawa, Kasraoui and Zeng using the transfer-matrix method. In this paper, we shall give bijective proofs of all the conjectures of Steingrímsson. Our basic idea is to encode ordered set partitions by a kind of path diagrams and explore the rich combinatorial properties of the latter structure. As a bonus of our approach, we derive two new σ-partition interpretations of the p,q-Stirling numbers of the second kind introduced by Wachs and White. We also discuss the connections with MacMahon's theorem on the equidistribution of the inversion number and major index on words and give a partition version of his result.     

Lattice path encodings in a combinatorial proof of a differential identity

We specify procedures by which ?ukasiewicz paths can encode combinatorial objects, such as involutions, partitions, and permutations. As application, we use these encoding procedures to give a combinatorial proof of the differential operator identity

     

Some sieves for partition theory

A Combinatorial lemma due to Zolnowsky is applied to partition theory in a new way. Several even-odd formulas are derived. A combinatorial proof of the Jacobi triple-product identity is given. Some sieve formulas are proved which relate to the Euler pentagonal number theorem and the Rogers-Ramanujan identities and their generalizations. A formula is given for the number of partitions of n into parts not congruent to 0, ±x (mod y).     

Bijective proofs using two-line matrix representations for partitions

In this paper, we present bijective proofs of several identities involving partitions by making use of a new way for representing partitions as two-line matrices. We also apply these ideas to give a combinatorial proof for an identity related to three-quadrant Ferrers graphs.     

An analogue of Euler''s identity and new combinatorial properties of n-colour compositions

An analogue of Euler's partition identity: “The number of partitions of a positive integer ν into odd parts equals the number of its partitions into distinct parts” is obtained for ordered partitions. The ideas developed are then used in obtaining several new combinatorial properties of the n-colour compositions introduced recently by the author.     

Congruences for broken 3-diamond and 7 dots bracelet partitions

Andrews and Paule introduced broken k-diamond partitions by using MacMahon’s partition analysis. Later, Fu found a generalization which he called k dots bracelet partitions. In this paper, with the aid of Farkas and Kra’s partition theorem and a p-dissection identity of f(?q), we derive many congruences for broken 3-diamond partitions and 7 dots bracelet partitions.     

A direct combinatorial proof of the Jacobi identity

An alternate form of the Jacobi identity is equivalent to the assertion that the number of partitions of a Gaussian integer r + si into an odd number of distinct non-zero Gaussian integers p + qi such that ip ? qi ≤ 1,p≥0,q≥0, is equal to the number of partitions into an even number of such integers, except when r and s are consecutive triangular numbers. A proof of this assertion is given, based on a dot diagram analogous to that used in Franklin's proof of Euler's theorem relating to the number of partitions of a natural integer into an odd and an even number of distinct parts.