Labeled Dyck Paths Research Articles

Some Consequences of the Valley Delta Conjectures

AbstractHaglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018) introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function $$\Delta '_{e_{n-k-1}} e_n$$ Δ e n - k - 1 ′ e n in terms of rise-decorated or valley-decorated labelled Dyck paths. While the rise version has been recently proved (D’Adderio and Mellit in Adv Math 402:108342, 2022; Blasiak et al. in A Proof of the Extended Delta Conjecture, arXiv:2102.08815, 2021), not much is known about the valley version. In this work, we prove the Schröder case of the valley Delta conjecture, the Schröder case of its square version (Iraci and Wyngaerd in Ann Combin 25(1):195–227, 2021), and the Catalan case of its extended version (Qiu and Wilson in J Combin Theory Ser A 175:105271, 2020). Furthermore, assuming the symmetry of (a refinement of) the combinatorial side of the extended valley Delta conjecture, we deduce also the Catalan case of its square version (Iraci and Wyngaerd 2021).

Volumes of Flow Polytopes Related to Caracol Graphs

Recently, Benedetti et al. introduced an Ehrhart-like polynomial associated to a graph. This polynomial is defined as the volume of a certain flow polytope related to a graph and has the property that the leading coefficient is the volume of the flow polytope of the original graph with net flow vector $(1,1,\dots,1)$. Benedetti et al. conjectured a formula for the Ehrhart-like polynomial of what they call a caracol graph. In this paper their conjecture is proved using constant term identities, labeled Dyck paths, and a cyclic lemma.

Phylogenetic Trees, Augmented Perfect Matchings, and a Thron-type Continued Fraction (T-fraction) for the Ward Polynomials

We find a Thron-type continued fraction (T-fraction) for the ordinary generating function of the Ward polynomials, as well as for some generalizations employing a large (indeed infinite) family of independent indeterminates. Our proof is based on a bijection between super-augmented perfect matchings and labeled Schröder paths, which generalizes Flajolet's bijection between perfect matchings and labeled Dyck paths.

Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application

In this paper, we study the problem of developing new combinatorial generation algorithms. The main purpose of our research is to derive and improve general methods for developing combinatorial generation algorithms. We present basic general methods for solving this task and consider one of these methods, which is based on AND/OR trees. This method is extended by using the mathematical apparatus of the theory of generating functions since it is one of the basic approaches in combinatorics (we propose to use the method of compositae for obtaining explicit expression of the coefficients of generating functions). As a result, we also apply this method and develop new ranking and unranking algorithms for the following combinatorial sets: permutations, permutations with ascents, combinations, Dyck paths with return steps, labeled Dyck paths with ascents on return steps. For each of them, we construct an AND/OR tree structure, find a bijection between the elements of the combinatorial set and the set of variants of the AND/OR tree, and develop algorithms for ranking and unranking the variants of the AND/OR tree.

From Dyck Paths to Standard Young Tableaux

We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength $n$ that is shown to be in bijection with the set of all SYT with $n$ boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some sense is dual to the known class of noncrossing partitions.

On random polynomials generated by a symmetric three-term recurrence relation

ABSTRACTWe investigate the sequence of random polynomials generated by the three-term recurrence relation , , with initial conditions , , assuming that is a sequence of positive i.i.d. random variables. is a sequence of orthogonal polynomials on the real line, and is the characteristic polynomial of a Jacobi matrix . We investigate the relation between the common distribution of the recurrence coefficients and two other distributions obtained as weak limits of the averaged empirical and spectral measures of . Our main result is a description of combinatorial relations between the moments of the aforementioned distributions in terms of certain classes of coloured planar trees. Our approach is combinatorial, and the starting point of the analysis is a formula of P. Flajolet for weight polynomials associated with labelled Dyck paths.

The Schröder case of the generalized Delta conjecture

We prove the Schröder case, i.e. the case 〈⋅,en−dhd〉, of the conjecture of Haglund et al. (2018) for ΔhmΔen−k−1′en in terms of decorated partially labelled Dyck paths, which we call generalized Delta conjecture. This result extends the Schröder case of the Delta conjecture proved in (D’Adderio, 2017), which in turn generalized the q,t-Schröder in Haglund (2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another combinatorial interpretation of the same polynomial in terms of a new bounce statistic. Moreover, we give two more interpretations of the same polynomial in terms of doubly decorated parallelogram polyominoes, extending some of the results in D’Adderio (2017), which in turn extended results in Aval et al., (2014). Also, we provide combinatorial bijections explaining some of the equivalences among these interpretations.

On Parking Functions and the Zeta Map in Types B, C and D

Let $\Phi$ be an irreducible crystallographic root system with Weyl group $W$, coroot lattice $\check{Q}$ and Coxeter number $h$. Recently the second named author defined a uniform $W$-isomorphism $\zeta$ between the finite torus $\check{Q}/(mh+1)\check{Q}$ and the set of non-nesting parking functions $\operatorname{Park}^{(m)}(\Phi)$. If $\Phi$ is of type $A_{n-1}$ and $m=1$ this map is equivalent to a map defined on labelled Dyck paths that arises in the study of the Hilbert series of the space of diagonal harmonics. In this paper we investigate the case $m=1$ for the other infinite families of root systems ($B_n$, $C_n$ and $D_n$). In each type we define models for the finite torus and for the set of non-nesting parking functions in terms of labelled lattice paths. The map $\zeta$ can then be viewed as a map between these combinatorial objects. Our work entails new bijections between (square) lattice paths and ballot paths.

A Decomposition of Parking Functions by Undesired Spaces

There is a well-known bijection between parking functions of a fixed length and maximal chains of the noncrossing partition lattice which we can use to associate to each set of parking functions a poset whose Hasse diagram is the union of the corresponding maximal chains. We introduce a decomposition of parking functions based on the largest number omitted and prove several theorems about the corresponding posets. In particular, they share properties with the noncrossing partition lattice such as local self-duality, a nice characterization of intervals, a readily computable Möbius function, and a symmetric chain decomposition. We also explore connections with order complexes, labeled Dyck paths, and rooted forests.

How to decompose a permutation into a pair of labeled Dyck paths by playing a game

We give a bijection between permutations of 1 , … , 2 n 1,\ldots ,2n and certain pairs of Dyck paths with labels on the down steps. The bijection arises from a game in which two players alternate selecting from a set of 2 n 2n items: the permutation encodes the players’ preference ordering of the items, and the Dyck paths encode the order in which items are selected under optimal play. We enumerate permutations by certain new statistics, AA inversions and BB inversions, which have natural interpretations in terms of the game. We derive identities such as \[ ∑ p ∏ i = 1 n q h i − 1 [ h i ] q = [ 1 ] q [ 3 ] q ⋯ [ 2 n − 1 ] q \sum _{p} \prod _{i=1}^n q^{h_i -1} [h_i]_q = [1]_q [3]_q \cdots [2n-1]_q \] where the sum is over all Dyck paths p p of length 2 n 2n , and h 1 , … , h n h_1,\ldots ,h_n are the heights of the down steps of p p .

New combinatorial formulations of the shuffle conjecture

The shuffle conjecture (due to Haglund, Haiman, Loehr, Remmel, and Ulyanov) provides a combinatorial formula for the Frobenius series of the diagonal harmonics module DHn, which is the symmetric function ∇(en). This formula is a sum over all labeled Dyck paths of terms built from combinatorial statistics called area, dinv, and IDes. We provide three new combinatorial formulations of the shuffle conjecture based on other statistics on labeled paths, parking functions, and related objects. Each such reformulation arises by introducing an appropriate new definition of the inverse descent set. Analogous results are proved for the higher-order shuffle conjecture involving ∇m(en). We also give new versions of some recently proposed combinatorial formulas for ∇(Cα) and ∇(s(k,1(n−k))), which translate expansions based on the dinv statistic into equivalent expansions based on Haglund's bounce statistic.

Dyck tableaux

We introduce and study new combinatorial objects called Dyck tableaux which may be seen as a variant of permutation tableaux. These objects appear in the combinatorial interpretation of the physical model PASEP (Partially Simple Asymmetric Exclusion Process). Dyck tableaux afford a simple recursive structure through the construction of an insertion algorithm. With this tool, we are able to describe statistics which are relevant in the PASEP model, in a more direct way than in previous works. Moreover, we give a new and natural link between permutations and certain labeled Dyck paths known as subdivided Laguerre histories.

Labeled Ballot Paths and the Springer Numbers

The Springer numbers are defined in connection with the irreducible root system of type $B_n$ and also arise as the generalized Euler and class numbers introduced by Shanks. Combinatorial interpretations of the Springer numbers have been found by Purtill in terms of Andre signed permutations, and by Arnol'd in terms of snakes of type $B_n$. We introduce the inversion code of a snake of type $B_n$ and establish a bijection between labeled ballot paths of length $n$ and snakes of type $B_n$. Moreover, we obtain the bivariate generating function for the number $B(n,k)$ of labeled ballot paths starting at $(0,0)$ and ending at $(n,k)$. Using our bijection, we find a statistic $\alpha$ such that the number of snakes $\pi$ of type $B_n$ with $\alpha(\pi)=k$ equals $B(n,k)$. We also show that our bijection specializes to a bijection between labeled Dyck paths of length $2n$ and alternating permutations on $[2n]$.

Nested Quantum Dyck Paths and (s )

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron-Garsia nabla operator. The formula involves nested labelled Dyck paths weighted by area and a suitable “diagonal inversion” statistic. Our model includes as special cases many previous conjectures connecting the nabla operator to quantum lattice paths. The combinatorics of the inverse Kostka matrix leads to an elementary proof of our proposed formula when q = 1. We also outline a possible approach for proving all the extant nabla conjectures that reduces everything to the construction of sign-reversing involutions on explicit collections of signed, weighted objects.

Indecomposable permutations, hypermaps and labeled Dyck paths

Hypermaps were introduced as an algebraic tool for the representation of embeddings of graphs on an orientable surface. Recently a bijection was given between hypermaps and indecomposable permutations; this sheds new light on the subject by connecting a hypermap to a simpler object. In this paper, a bijection between indecomposable permutations and labeled Dyck paths is proposed, from which a few enumerative results concerning hypermaps and maps follow. We obtain for instance an inductive formula for the number of hypermaps with n darts, p vertices and q hyperedges; the latter is also the number of indecomposable permutations of S n with p cycles and q left-to-right maxima. The distribution of these parameters among all permutations is also considered.

Square $\boldsymbol {q,t}$-lattice paths and $\boldsymbol {\nabla (p_n)}$

The combinatorial q,t-Catalan numbers are weighted sums of Dyck paths introduced by J. Haglund and studied extensively by Haglund, Haiman, Garsia, Loehr, and others. The q,t-Catalan numbers, besides having many subtle combinatorial properties, are intimately connected to symmetric functions, algebraic geometry, and Macdonald polynomials. In particular, the n'th q,t-Catalan number is the for the module of diagonal harmonic alternants in 2n variables; it is also the coefficient of s 1 n in the Schur expansion of ∇(e n ). Using q,t-analogues of labelled Dyck paths, Haglund et al. have proposed combinatorial conjectures for the monomial expansion of ∇(e n ) and the of the diagonal harmonics modules. This article extends the combinatorial constructions of Haglund et al. to the case of lattice paths contained in squares. We define and study several q,t-analogues of these lattice paths, proving combinatorial facts that closely parallel corresponding results for the q,t-Catalan polynomials. We also conjecture.an interpretation of our combinatorial polynomials in terms of the nabla operator. In particular, we conjecture combinatorial formulas for the monomial expansion of ∇(p n ), the Hilbert series (∇(p n ), h 1 n), and the sign character (∇(p n ),s 1 n).

Conjectured Combinatorial Models for the Hilbert Series of Generalized Diagonal Harmonics Modules

Haglund and Loehr previously conjectured two equivalent combinatorial formulas for the Hilbert series of the Garsia-Haiman diagonal harmonics modules. These formulas involve weighted sums of labelled Dyck paths (or parking functions) relative to suitable statistics. This article introduces a third combinatorial formula that is shown to be equivalent to the first two. We show that the four statistics on labelled Dyck paths appearing in these formulas all have the same univariate distribution, which settles an earlier question of Haglund and Loehr. We then introduce analogous statistics on other collections of labelled lattice paths contained in trapezoids. We obtain a fermionic formula for the generating function for these statistics. We give bijective proofs of the equivalence of several forms of this generating function. These bijections imply that all the new statistics have the same univariate distribution. Using these new statistics, we conjecture combinatorial formulas for the Hilbert series of certain generalizations of the diagonal harmonics modules.

Théorie combinatoire des T-fractions et approximants de Padé en deux points

We present a combinatorial interpretation for Thron's continued fractions (T-fractions) in terms of labelled Dyck paths. The links between convergents of T-fractions and two points Padé approximants arise naturally. Then we show a new bijection between permutations and some combinatorial histories, allowing us to compute the generating function of a trivariate statistics on permutations. Finally, a strange application of the duality of T-fractions is given.