Bijective Proof Research Articles

Combinatorial proofs and refinements of three partition theorems of Andrews

In his recent work Andrews revisited two-color partitions with certain restrictions on the differences between consecutive parts, and he established three theorems linking these two-color partitions with more familiar kinds of partitions. In this note we provide bijective proofs as well as refinements of those three theorems of Andrews. Our refinements take into account the numbers of parts in each of the two colors.

Partial Rank Symmetry of Distributive Lattices for Fences

Associated with any composition \(\beta =(a,b,\ldots )\) is a corresponding fence poset \(F(\beta )\) whose covering relations are $$\begin{aligned} x_1\lhd x_2 \lhd \ldots \lhd x_{a+1}\rhd x_{a+2}\rhd \ldots \rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots . \end{aligned}$$The distributive lattice \(L(\beta )\) of all lower order ideals of \(F(\beta )\) is important in the theory of cluster algebras. In addition, its rank generating function \(r(q;\beta )\) is used to define q-analogues of rational numbers. Kantarcı Oğuz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a conjecture of McConville, Smyth, and Sagan, which in turn implies a previous conjecture of Morier-Genoud and Ovsienko that \(r(q;\beta )\) is unimodal. We show that, when \(\beta \) has an odd number of parts, then the polynomial is also partially symmetric: the number of ideals of \(F(\beta )\) of size k equals the number of filters of size k, when k is below a certain value. Our proof is completely bijective. Kantarcı Oğuz and Ravichandran also introduced a circular version of fences and proved, using algebraic techniques, that the distributive lattice for such a poset is rank symmetric. We give a bijective proof of this result, as well. We end with some questions and conjectures raised by this work.

Bijective Proofs of Formulas with (-1) n

Bijective Proofs of Formulas with (-1) ⁿ

On Generalizations of Theorems of MacMahon and Subbarao

In this paper, we consider various theorems of P.A. MacMahon and M.V. Subbarao. For a non-negative integer n, MacMahon proved that the number of partitions of n wherein parts have multiplicity greater than 1 is equal to the number of partitions of n in which odd parts are congruent to 3 modulo 6. We give a new bijective proof for this theorem and its generalization, which consequently provides a new proof of Andrews’ extension of the theorem. Considering Subbarao’s finitization of Andrews’ extension, we generalize this result of Subbarao. Our generalization is based on Glaisher’s extension of Euler’s mapping for odd-distinct partitions and as a result, a bijection given by Sellers and Fu is also extended. Unlike in the case of Sellers and Fu where two residue classes are fixed, ours takes into consideration all possible residues. Furthermore, some arithmetic properties of related partition functions are derived.

On a Conjecture Concerning Shuffle-Compatible Permutation Statistics

The notion of shuffle-compatible permutation statistics was implicit in Stanley's work on P-partitions and was first explicitly studied by Gessel and Zhuang. The aim of this paper is to prove that the triple ${\rm (udr, pk, des)}$ is shuffle-compatible as conjectured by Gessel and Zhuang, where ${\rm udr}$ denotes the number of up-down runs, ${\rm pk}$ denotes the peak number, and ${\rm des}$ denotes the descent number. This is accomplished by establishing an ${\rm (udr, pk, des)}$-preserving bijection in the spirit of Baker-Jarvis and Sagan's bijective proofs of the shuffle compatibility property of permutation statistics.

On the combinatorics of derangements and related permutations

A derangement is a permutation in which no entry is at its original position. The number of derangements of [n] is called the “derangement number” or “de Montmort number”, and is denoted by Dn. The sequence {Dn} enumerates, in addition to the number of derangements, many other permutations under various constraints. In this paper, we explore the connections between these combinatorial objects and provide bijective proofs. Some related enumerative problems are also mentioned.

Beyond Göllnitz' theorem II: Arbitrarily many primary colors

In 2003, Alladi, Andrews and Berkovich proved a four parameter partition identity lying beyond a celebrated identity of Göllnitz. Since then it has been an open problem to extend their work to five or more parameters. In part I of this pair of papers, we took a first step in this direction by giving a bijective proof of a reformulation of their result. We introduced forbidden patterns, bijectively proved a ten-colored partition identity, and then related, by another bijection, our identity to the Alladi-Andrews-Berkovich identity.In this second paper, we state and bijectively prove an n(n+1)2-colored partition identity beyond Göllnitz' theorem for any number n of primary colors, along with the full set of the n(n−1)2 secondary colors as the product of two distinct primary colors, generalizing the identity proved in the first paper. Like the ten-colored partitions, our family of n(n+1)2-colored partitions satisfy some simple minimal difference conditions while avoiding forbidden patterns. Furthermore, the n(n+1)2-colored partitions have some remarkable properties, as they can be uniquely represented by oriented rooted forests which record the steps of the bijection.

Bijections for ranked tree-child networks

The class of ranked tree-child networks, tree-child networks arising from an evolution process with a fixed embedding into the plane, has recently been introduced by Bienvenu, Lambert, and Steel. These authors derived counting results for this class. In this note, we will give bijective proofs of three of their results. Two of our bijections answer questions raised in their paper.

A bijective proof of Kohnert's rule for Schubert polynomials

Kohnert proposed the first monomial positive formula for Schubert polynomials as the generating polynomial for certain unit cell diagrams obtained from the Rothe diagram of a permutation. Billey, Jockusch and Stanley gave the first proven formula for Schubert polynomials as the generating polynomial for compatible sequences of reduced words of a permutation. In this paper, we give an explicit bijection between these two models, thereby definitively proving Kohnert's rule for Schubert polynomials.

Involutions under Bruhat order and labeled Motzkin paths

In this note, we introduce a statistic on Motzkin paths that describes the rank generating function of Bruhat order for involutions. Our proof relies on a bijection introduced by P. Biane from permutations to certain labeled Motzkin paths and a recently introduced interpretation of this rank generating function in terms of visible inversions. By restricting our identity to fixed-point-free (FPF) involutions, we recover an identity due to L. Billera, L. Levine and K. Mészáros with a previous bijective proof by M. Watson. Our work sheds new light on the Ethiopian dinner game.

A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions

We give a new characterization of the vertical-strip LLT polynomials GP(x;q) as the unique family of symmetric functions that satisfy certain combinatorial relations. This characterization is then used to prove an explicit combinatorial expansion of vertical-strip LLT polynomials in terms of elementary symmetric functions. Such formulas were conjectured independently by A. Garsia et al. and the first named author, and are governed by the combinatorics of orientations of unit-interval graphs. The obtained expansion is manifestly positive if q is replaced by q+1, thus recovering a recent result of M. D'Adderio. Our results are based on linear relations among LLT polynomials that arise in the work of D'Adderio, and of E. Carlsson and A. Mellit. To some extent these relations are given new bijective proofs using colorings of unit-interval graphs. As a bonus we obtain a new characterization of chromatic quasisymmetric functions of unit-interval graphs.

Partitions with k sizes from a set

Let $n$ and $k$ be two positive integers and let $A$ be a set of positive integers. We define $t_A(n,k)$ to be the number of partitions of $n$ with exactly $k$ sizes and parts in $A$. As an implication of a variant of Newton's product-sum identities we present a generating function for $t_A(n,k)$. Subsequently, we obtain a recurrence relation for $t_A(n,k)$ and a divisor-sum expression for $t_A(n,2)$. Also, we present a bijective proof for the latter expression.

Orthosymplectic Cauchy Identities

We give bijective proofs of orthosymplectic analogues of the Cauchy identity and dual Cauchy identity for orthosymplectic Schur functions. To do so, we present two insertion algorithms; these are orthosymplectic versions of Berele’s symplectic insertion algorithms, which were used by Sundaram to give bijective proofs of Cauchy identities for symplectic Schur functions.

Bijections for Faces of the Shi and Catalan Arrangements

In 1986, Shi derived the famous formula $(n+1)^{n-1}$ for the number of regions of the Shi arrangement, a hyperplane arrangement in ${R}^n$. There are at least two different bijective explanations of this formula, one by Pak and Stanley, another by Athanasiadis and Linusson. In 1996, Athanasiadis used the finite field method to derive a formula for the number of $k$-dimensional faces of the Shi arrangement for any $k$. Until now, the formula of Athanasiadis did not have a bijective explanation. In this paper, we extend a bijection for regions defined by Bernardi to obtain a bijection between the $k$-dimensional faces of the Shi arrangement for any $k$ and a set of decorated binary trees. Furthermore, we show how these trees can be converted to a simple set of functions of the form $f: [n-1] \to [n+1]$ together with a marked subset of $\text{Im}(f)$. This correspondence gives the first bijective proof of the formula of Athanasiadis. In the process, we also obtain a bijection and counting formula for the faces of the Catalan arrangement. All of our results generalize to both extended arrangements.

On Andrews’ Partitions with Parts Separated by Parity

In this paper, we present a generalization of one of the theorems in Partitions with parts separated by parity introduced by George E. Andrews, and give its bijective proof. Further variations of related partition functions are studied resulting in a number of interesting identities.

Edges in Fibonacci Cubes, Lucas Cubes and Complements

The Fibonacci cube of dimension n, denoted as \(\Gamma _n\), is the subgraph of the hypercube induced by vertices with no consecutive 1s. The Lucas cube \(\Lambda _n\) is the cyclic version of \(\Gamma _n\). The irregularity of a graph G is the sum of \(|d(x)-d(y)|\) over all edges \(\{x,y\}\) of G. In two recent papers based on the recursive structure of \(\Gamma _n\), it is proved that the irregularity of \(\Gamma _n\) and \(\Lambda _n\) is two times the number of edges of \(\Gamma _{n-1}\) and 2n times the number of vertices of \(\Gamma _{n-4}\), respectively. Using an interpretation of the irregularity in terms of couples of incident edges of a special kind, we give a bijective proof of both results. For these two graphs, we deduce also a constant time algorithm for computing the imbalance of an edge. In the last section using the same approach, we determine the number of edges and the sequence of degrees of the cube complement of \(\Gamma _n\).

Bijective Proofs of Proper Coloring Theorems

The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynomial case, and the bijection for broken circuits is just a restriction of the one for all spanning subgraphs.

Enumeration of Lattice paths with infinite types of steps and the Chung-Feller property

In this paper we consider the lattice paths that go from the origin to a point on the right half of the plane with step set S={Ui=(1+i,1−i)|i≥0}⋃{Vi=(i,−i)|i≥1} such that the step Ui is assigned with weight ui and the step Vi is assigned with weight vi for i≥1 except that the step U0=(1,1) is assigned with weight 1. Let G(n,k) be the set of all lattice paths ending at the point (n,2k−n) and Gn,k=|G(n,k)|, and let U(n,k) (resp. V(n,k)) be the set of all lattice paths (resp. nonnegative lattice paths) ending at (2n−k,k) and Un,k=|U(n,k)| (resp. Vn,k=|V(n,k)|). We will show that (Gn,k)n,k∈N, (Un,k)n,k∈N and (Vn,k)n,k∈N are all Riordan arrays. Correlations between these Riordan arrays are studied. Consequently, a new Chung-Feller type property is obtained, and the bijective proof is provided. We also list numerous interesting examples.

Flip-sort and combinatorial aspects of pop-stack sorting

Flip-sort is a natural sorting procedure which raises fascinating combinatorial questions. It finds its roots in the seminal work of Knuth on stack-based sorting algorithms and leads to many links with permutation patterns. We present several structural, enumerative, and algorithmic results on permutations that need few (resp. many) iterations of this procedure to be sorted. In particular, we give the shape of the permutations after one iteration, and characterize several families of permutations related to the best and worst cases of flip-sort. En passant, we also give some links between pop-stack sorting, automata, and lattice paths, and introduce several tactics of bijective proofs which have their own interest.

A generalization of Edelman–Greene insertion for Schubert polynomials

Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman--Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial variation, but here we demonstrate how to extract explicit formulas from that machinery which may be applied to other positivity problems as well.