Bijective Proof Research Articles

Lakshmibai-Seshadri Paths and Non-Symmetric Cauchy Identity

We give a simple crystal theoretic interpretation of the Lascoux’s expansion of a non-symmetric Cauchy kernel ${\prod }_{i+ j \leq n + 1}(1-x_{i}y_{j})^{-1}$ , which is given in terms of Demazure characters and atoms. We give a bijective proof of the non-symmetric Cauchy identity using the crystal of Lakshmibai-Seshadri paths, and extend it to the case of continuous crystals.

A Bijective Proof of a Derangement Recurrence

A Bijective Proof of a Derangement Recurrence

A trinity of duality: Non-separable planar maps, β(1,0)-trees and synchronized intervals

The dual of a map is a fundamental construction on combinatorial maps, but many other combinatorial objects also possess their notion of duality. For instance, the Tamari lattice is isomorphic to its order dual, which induces an involution on the set of so-called “synchronized intervals” introduced by Préville-Ratelle and the present author. Another example is the class of β(1,0)-trees, which has a mysterious involution h proposed by Claesson, Kitaev and Steingrímsson (2009). These two classes of combinatorial objects are all in bijection with the class of non-separable planar maps, which is closed under map duality. In this article, we show that we can identify the notions of duality in these three classes using previously known natural bijections, which leads to a bijective proof of a result from Kitaev and de Mier (2013).

Generalizing inplace multiplicity identities for integer compositions

In a recent paper, the authors gave two new identities for compositions, or ordered partitions, of integers. These identities were based on closely-related integer partition functions which have recently been studied. In the process, we also extensively generalized both of these identities. Since then, we asked whether one could generalize one of these results even further by considering compositions in which certain parts could come from t kinds (rather than just two kinds, which was the crux of the original result). In this paper, we provide such a generalization. A straightforward bijective proof is given and generating functions are provided for each of the types of compositions which arise. We close by briefly mentioning some arithmetic properties satisfied by the functions which count such compositions.

A bijective proof of the hook-length formula for skew shapes

Recently, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. In this extended abstract, we present a simple bijection that proves an equivalent recursive version of Naruse's result, in the same way that the celebrated hook-walk proof due to Green, Nijenhuis and Wilf gives a bijective (or probabilistic) proof of the hook-length formula for ordinary shapes. In particular, we also give a new bijective proof of the classical hook-length formula, quite different from the known proofs.

Two infinite families of terminating binomial sums

We present a family of identities including both binomial coefficients and a power of a natural number $$m \ge 2$$ . We find a combinatorial interpretation of these identities, which provides bijective proof. Dual alternating sign identities are also presented.

New Combinatorial Formulas for Cluster Monomials of Type $A$ Quivers

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type $A$ cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the $T$-paths and of the perfect matchings in a snake diagram. For cluster variables of a type $A$ cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.

Partition identities and quiver representations

We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy’s Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke’s identity in the case of quivers \({\mathcal {Q}}\) of Dynkin type A. Our identity is stated in terms of the lacing diagrams of S. Abeasis–A. Del Fra, which parameterize orbits of the representation space of \({\mathcal {Q}}\) for a fixed dimension vector.

Asymptotic and Exact Results on the Complexity of the Novelli-Pak-Stoyanovskii Algorithm

The Novelli-Pak-Stoyanovskii algorithm is a sorting algorithm for Young tableaux of a fixed shape that was originally devised to give a bijective proof of the hook-length formula. We obtain new asymptotic results on the average case and worst case complexity of this algorithm as the underlying shape tends to a fixed limit curve. Furthermore, using the summation package Sigma we prove an exact formula for the average case complexity when the underlying shape consists of only two rows. We thereby answer questions posed by Krattenthaler and Müller.

Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan Numbers

The d-dimensional Catalan numbers form a well-known sequence of numbers which count balanced bracket expressions over an alphabet of size d. In this paper, we introduce and study what we call d-dimensional prime Catalan numbers, a sequence of numbers which count only a very specific subset of indecomposable balanced bracket expressions. These numbers were encountered during the investigation of what we call trapezoidal diagrams of geometric graphs, such as triangulations or crossing-free perfect matchings. In essence, such a diagram is obtained by augmenting the geometric graph in question with its trapezoidal decomposition, and then forgetting about the precise coordinates of individual vertices while preserving the vertical visibility relations between vertices and segments. We note that trapezoidal diagrams of triangulations are closely related to abstract upward triangulations. We study the numbers of such diagrams in the cases of (i) perfect matchings and (ii) triangulations. We give bijective proofs which establish relations with 3-dimensional (prime) Catalan numbers. This allows us to determine the corresponding exponential growth rates exactly as (i) $$5.196^n$$ and (ii) $$23.459^n$$ (bases are rounded to three decimal places). Finally, we give exponential lower bounds for the maximum number of embeddings that a trapezoidal diagram can have on any given point set.

On partitions with initial repetitions

We consider George Andrews’ fundamental theorem on partitions with initial repetitions and obtain some partition identities and parity results. A simplified, diagram-free, version of William Keith’s bijective proof of the theorem is presented. Lastly, we obtain extensions and variations of the theorem using a class of Rogers–Ramanujan-type identities for n-color partitions studied by A.K. Agarwal.

A New Bijective Proof of Babson and Steingrímsson's Conjecture

Babson and Steingrímsson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $\pi$, let $des(\pi)$ denote the descent number of $\pi$ and $maj(\pi)$ denote the major index of $\pi$. Babson and Steingrímsson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.

A permutation code preserving a double Eulerian bistatistic

In 1977 Foata proved bijectively, among other things, that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. In 2013 Foata’s result was rediscovered by Visontai as a conjecture, and then reproved by Aas in 2014.In this paper, we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two 5-tuples of set-valued statistics on the set of permutations and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give another bijective proof of Foata’s result.

A bijective proof of a q-analogue of the sum of cubes using overpartitions

A bijective proof of a q-analogue of the sum of cubes using overpartitions

A Bijective Proof of Vershik’s Relations for the Kostka Numbers

We turn the tableau insertion algorithm for semistandard tableaux to a sequence called a buming route. We also define a reverse bumping route by the reverse insertion algorithm. These sequences are used to give a bijective proof of Vershik's relations for the Kostka numbers.

Cycles in the Graph of Overlapping Permutations Avoiding Barred Patterns

As a variation of De Bruijn graphs on strings of symbols, the graph of overlapping permutations has a directed edge $\pi(1)\pi(2)\ldots \pi(n+1)$ from the standardization of $\pi(1)\pi(2)\ldots \pi(n)$ to the standardization of $\pi(2)\pi(3)\ldots \pi(n+1)$. In this paper, we consider the enumeration of $d$-cycles in the subgraph of overlapping $(231, 4\bar{1}32)$-avoiding permutations. To this end, we introduce the notions of marked Motzkin paths and marked Riordan paths, where a marked Motzkin (resp. Riordan) path is a Motzkin (resp. Riordan) path in which exactly one step before the leftmost return point is marked. We show that the number of closed walks of length $d$ in the subgraph of overlapping $(231, 4\bar{1}32)$-avoiding permutations are closely related to the number of marked Motzkin paths and that of marked Riordan paths. By establishing bijections, we get the enumerations of marked Motzkin paths and marked Riordan paths. As a corollary, we provide bijective proofs of two identities involving Catalan numbers in answer to the problem posed by Ehrenborg, Kitaev and Steingrímsson. Moreover, we get the enumerations of $(231, 4\bar{1}32)$-avoiding affine permutations and $(312, 32\bar{4}1)$-avoiding affine permutations.

Variants of the RSK algorithm adapted to combinatorial Macdonald polynomials

We introduce variations of the Robinson–Schensted correspondence parametrized by positive integers p. Each variation gives a bijection between permutations and pairs of standard tableaux of the same shape. In addition to sharing many of the properties of the classical Schensted algorithm, the new algorithms are designed to be compatible with certain permutation statistics introduced by Haglund in the study of Macdonald polynomials. In particular, these algorithms provide an elementary bijective proof converting Haglund's combinatorial formula for Macdonald polynomials to an explicit combinatorial Schur expansion of Macdonald polynomials indexed by partitions μ satisfying μ1≤3 and μ2≤2. We challenge the research community to extend this RSK-based approach to more general classes of partitions.

Restricted Composition Deletion Correcting Codes

We investigate deletion correcting codes and restricted composition codes in particular. Restricted composition codes generalize codes on permutations and multipermutations. We use graph theoretic methods to characterize codes, establish bounds on code size, and describe constructions. The substring partial order has a well-known property. For any string, the number of superstrings of a particular length depends only on the length of the original string. We generalize this property to take compositions into account. For any string, the number of superstrings of a particular composition depends only on the composition of the original string. We present a bijective proof of this fact. We apply this result to obtain a lower bound on the size of restricted composition codes. We obtain an upper bound by analyzing deletion errors when the composition of deleted symbols is restricted to a particular worst case composition. We construct binary restricted composition single-deletion correcting codes and show that they are asymptotically optimal and form an optimal coloring. There is a natural distance on compositions that provides a lower bound on deletion distance. Unrestricted deletion correcting codes can be constructed from the union of restricted composition codes as long as the set of compositions used themselves form a code. The nonbinary single-deletion correcting codes constructed by Tenengolts are a special case of our method.

Face Rings of Cycles, Associahedra, and Standard Young Tableaux

We show that $J_n$, the Stanley-Reisner ideal of the $n$-cycle, has a free resolution supported on the $(n-3)$-dimensional simplicial associahedron $A_n$. This resolution is not minimal for $n \geq 6$; in this case the Betti numbers of $J_n$ are strictly smaller than the $f$-vector of $A_n$. We show that in fact the Betti numbers $\beta_{d}$ of $J_n$ are in bijection with the number of standard Young tableaux of shape $(d+1, 2, 1^{n-d-3})$. This complements the fact that the number of $(d-1)$-dimensional faces of $A_n$ are given by the number of standard Young tableaux of (super)shape $(d+1, d+1, 1^{n-d-3})$; a bijective proof of this result was first provided by Stanley. An application of discrete Morse theory yields a cellular resolution of $J_n$ that we show is minimal at the first syzygy. We furthermore exhibit a simple involution on the set of associahedron tableaux with fixed points given by the Betti tableaux, suggesting a Morse matching and in particular a poset structure on these objects.

Avoiding Monotone Chains in Fillings of Layer Polyominoes

In this paper we give simple bijective proofs that the number of fillings of layer polyominoes with no northeast chains is the same as the number with no southeast chains. We consider 01-fillings and \({{\mathbb{N}}}\)-fillings and prove the results for both strong chains where the smallest rectangle containing the chain is also in the polyomino, and for regular chains where only the corners of the smallest rectangle containing the chain are required to be in the polyomino.