In The Delta Conjecture (arxiv:1509.07058), Haglund, Remmel and Wilson introduced a four variable $q,t,z,w$ Catalan polynomial, so named because the specialization of this polynomial at the values $(q,t,z,w) = (1,1,0,0)$ is equal to the Catalan number $\frac{1}{n+1}\binom{2n}{n}$. We prove the compositional version of this conjecture (which implies the non-compositional version) that states that the coefficient of $s_{r,1^{n-r}}$ in the expression $Δ_{h_\ell} \nabla C_α$ is equal to a weighted sum over decorated Dyck paths.
We give conjectures on the form of families of integer sequences whose Hankel transforms are, respectively, $(α, β)$ Somos $4$ sequences, $(α, 0, γ)$ Somos $6$ sequences, and $(α, β, γ, δ)$ Somos $8$ sequences, for particular values of $α$, $β$, $γ$, $δ$ which we describe. The sequences involved can be described in terms of the application of certain stretched Riordan arrays to the Catalan numbers, accompanied by a (sequence) Hankel transform. The combination of Riordan array and the Catalan numbers results from the study of certain generalized Jacobi continued fractions, based on the Counting…
Let $\{P_n\}_{n\geq 0}$ denote the Catalan-Larcombe-French sequence, which naturally came up from the series expansion of the complete elliptic integral of the first kind. In this paper, we prove the strict log-concavity of the sequence $\{\sqrt[n]{P_n}\}_{n\geq 1}$, which was originally conjectured by Sun. We also obtain the strict log-concavity of the sequence $\{\sqrt[n]{V_n}\}_{n\geq 1}$, where $\{V_n\}_{n\geq 0}$ is the Fennessey-Larcombe-French sequence arising in the series expansion of the complete elliptic integral of the second kind.
We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana, Ohtsuka, and Romero [Discrete Math. 340 (2017), 2388--2397]. The first one is proved by using Zeilberger's algorithm, and the second one is proved by establishing its $q$-analogue.
The set of dominant regions of the $k$-Catalan arrangement of a crystallographic root system $Φ$ is a well-studied object enumerated by the Fuß-Catalan number $Cat^{(k)}(Φ)$. It is natural to refine this enumeration by considering floors and ceilings of dominant regions. A conjecture of Armstrong states that counting dominant regions by their number of floors of a certain height gives the same distribution as counting dominant regions by their number of ceilings of the same height. We prove this conjecture using a bijection that provides even more refined enumerative information.
While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and conjectures) are presented on this splitting of the squares that contribute to a refined understanding of the Kronecker squares. Furthermore, connections to 2-modular decomposition numbers, Catalan combinatorics, and to the Saxl conjecture are discussed which further motivate the study of these splittings.
A finitization of the Catalan numbers $ C_n $ can be defined as Euler characteristics of an algebraic structure. We conjecture the existence of a $ q $-deformed version of such structure, and provide evidence for the first two non-trivial cases. This $ q $-deformed version will likely categorify series arise from identities of the Roger-Ramanujan type.
A polynomial $A(q)=\sum_{i=0}^n a_iq^i$ is said to be unimodal if $a_0\le a_1\le \cdots \le a_k\ge a_{k+1} \ge \cdots \ge a_n$. We investigate the unimodality of rational $q$-Catalan polynomials, which is defined to be $C_{m,n}(q)= \frac{1}{[n+m]} \left[ m+n \atop n\right]$ for a coprime pair of positive integers $(m,n)$. We conjecture that they are unimodal with respect to parity, or equivalently, $(1+q)C_{m+n}(q)$ is unimodal. By using generating functions and the constant term method, we verify our conjecture for $m\le 5$ in a straightforward way.
Let $W$ be an irreducible real reflection group. Armstrong, Reiner, and the author presented a model for parking functions attached to W and made three increasingly strong conjectures about these objects. The author generalized these objects and conjectures to the Fuss-Catalan level of generality. Even the weakest of these conjectures would imply a collection of facts in Coxeter-Catalan theory which are at present understood only in a case-by-case fashion. We prove that when $W$ belongs to any the infinite families ABCDI, the strongest of these conjectures is generically true.
A beautiful recent conjecture of D. Armstrong predicts the average size of a partition that is simultaneously an $s$-core and a $t$-core, where $s$ and $t$ are coprime. Our goal is to prove this conjecture when $t=s+1$. These simultaneous $(s,s+1)$-core partitions, which are enumerated by Catalan numbers, have average size $\binom{s+1}{3}/2$.
We prove the following conjecture of Zeilberger. Denoting by $C_n$ the Catalan number, define inductively $A_n$ by $(-1)^{n-1}A_n=C_n+\sum_{j=1}^{n-1} (-1)^{j} \binom{2n-1}{2j-1} A_j \,C_{n-j}$ and $a_n=2A_n/C_n$. Then $a_n$ (hence $A_n$) is a positive integer.
We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.
In type A, the q,t-Fuss -Catalan numbers can be defined as a bigraded Hilbert series of a module associated to the symmetric group S_n. We generalize this construction to (finite) complex reflection groups and exhibit some nice conjectured algebraic and combinatorial properties of these polynomials in q and t. Finally, we present an idea how these polynomials could be related to some graded Hilbert series of modules arising in the context of rational Cherednik algebras. This is work in progress.
We prove that graded $k$-Schur functions are $G$-equivariant Euler characteristics of vector bundles on the flag variety, settling a conjecture of Chen-Haiman. We expose a new miraculous shift invariance property of the graded $k$-Schur functions and resolve the Schur positivity and $k$-branching conjectures in the strongest possible terms by providing direct combinatorial formulas using strong marked tableaux.
We give a simple recursion which computes the triply graded Khovanov-Rozansky homology of several infinite families of knots and links, including the $(n,nm\pm 1)$ and $(n,nm)$ torus links for $n,m\geq 1$. We interpret our results in terms of Catalan combinatorics, proving a conjecture of Gorsky's. Our computations agree with predictions coming from Hilbert schemes and rational DAHA, which also proves the Gorsky-Oblomkov-Rasmussen-Shende conjectures in these cases. Additionally, our results suggest a topological interpretation of the symmetric functions which appear in the context of the $m$-s…
We observe that for a and b relatively prime, the "abacus construction" identifies the set of simultaneous (a,b)-core partitions with lattice points in a rational simplex. Furthermore, many statistics on (a,b)-cores are piecewise polynomial functions on this simplex. We apply these results to rational Catalan combinatorics. Using Ehrhart theory, we reprove Anderson's theorem that there are (a+b-1)!/a!b! simultaneous (a,b)-cores, and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an (a,b)-core is (a+b+1)(a-1)(b-1)/24. Our methods also give new derivations …
In [The Delta Conjecture, Trans. Amer. Math. Soc., to appear] Haglund, Remmel, Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This operator, defined in terms of its action on the modified Macdonald basis, has played a role in work of Garsia and Haiman on diagonal harmonics, the Hilbert scheme, and Macdonald polynomials [A. M. Garsia and M. Haiman. A remarkable $q,t$-Catalan sequence and $q$-Lagrange inversion, J. Algebraic Combin. 5 (1996), 191--244], [M. Haiman, Vanishing theorems and char…
We conjecture two combinatorial interpretations for the symmetric function $Δ_{e_k} e_n$, where $Δ_f$ is an eigenoperator for the modified Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both interpretations can be seen as generalizations of the Shuffle Conjecture of Haglund, Haiman, Remmel, Loehr, and Ulyanov, which was proved recently by Carlsson and Mellit. We show how previous work of the third author on Tesler matrices and ordered set partitions can be used to verify several cases of our conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to …
We make a broad conjecture about the $k$-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which address the $k$-Schur expansion of (1) Hall-Littlewood polynomials, proving the $q=0$ case of the strengthened Macdonald positivity conjecture of Lapointe, Lascoux, and Morse; (2) the product of a Schur function and a $k$-Schur function when the indexing partitions concatenate to a partition, describing a class of Gromov-Witten invariants for the quantum coh…
This (partly expository) paper originated from the study of Hankel determinants of convolution powers of Catalan numbers and of Narayana polynomials. This led to some Hankel determinants of signed Catalan numbers whose values are multiples of Fibonacci numbers and to some Hankel determinants of signed central binomial coefficients whose values are multiples of Lucas numbers. Most proofs are computational but we also include a combinatorial one due to Christian Krattenthaler. Finally we formulate some conjectures.
An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that (2n)- and (2mn+1)-core partitions correspond naturally to dominant alcoves in the …
We define a module that is an extension of the diagonal harmonics and whose graded Frobenius characteristic is conjectured to be the symmetric function expression which appears in `the Delta conjecture' of Haglund, Remmel and Wilson [arXiv:1509.07058].
Inspired by [Qiu, Wilson 2019] and [D'Adderio, Iraci, Vanden Wyngaerd 2019 - Delta Square], we formulate a generalised Delta square conjecture (valley version). Furthermore, we use similar techniques as in [Haglund, Sergel 2019] to obtain a schedule formula for the combinatorics of our conjecture. We then use this formula to prove that the (generalised) valley version of the Delta conjecture implies our (generalised) valley version of the Delta square conjecture. This implication broadens the argument in [Sergel 2016], relying on the formulation of the touching version in terms of the $Θ_f$ op…
We prove the cases q=0 and t=0 of the generalized Delta conjecture of Haglund, Remmel and Wilson involving the symmetric function $Δ_{h_m}Δ_{e_{n-k-1}}'e_n$. Our theorem generalizes recent results by Garsia, Haglund, Remmel and Yoo. This proves also the case q=0 of our recent generalized Delta square conjecture.
A conjecture of Chunwei Song on a limiting case of the q,t-Schröder theorem is proved combinatorially. The proof matches pairs of tableaux to Catalan words in a manner that preserves differences in the maj statistic.
The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture in the field of diagonal harmonics. In this paper we give evidence for the Delta Conjecture by proving a pair of conjectures of Wilson and Haglund-Remmel-Wilson which give equidistribution results for statistics related to inversion count and major index on objects related to ordered set partitions. Our results generalize the famous result of MacMahon that major index and inversion number share the same distribution on permutations.
We prove a compositional refinement of the Delta conjecture (rise version) of Haglund, Remmel and Wilson (2018) for $Δ_{e_{n-k-1}}'e_n$ which was stated by D'Adderio, Iraci and Vanden Wyngaerd (2020) in terms of Theta operators.
In this paper we define two infinite families of graphs called C-$δ$ graphs and $δ$- graph and prove that $δ$-graphs satisfy $δ$ conjecture. Also we introduce a family of C-$δ$ graphs from which we can identify $δ$ graphs as their complements. Finally we give a list of C-$δ$ graphs and the relationship with their minimum semidefinite rank.
The \emph{$q,t$-Catalan numbers} $C_n(q,t)$ are polynomials in $q$ and $t$ that reduce to the ordinary Catalan numbers when $q=t=1$. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for $C_n(q,t)$ as weighted sums of Dyck paths (or equivalently, integer partitions contained in a staircase shape). This paper undertakes a combinatorial investigation of the joint symmetry property $C_n(q,t)=C_n(t,q)$. We conjecture some structural decompositions of Dyck objects into "mutually op…
In [Haglund, Remmel, Wilson 2018] the authors state two versions of the so called Delta conjecture, the rise version and the valley version. Of the former, they also give a more general statement in which zero labels are also allowed. In [Qiu, Wilson 2020], the corresponding generalisation of the valley version is also formulated. In [D'Adderio, Iraci, Vanden Wyngaerd 2020], the authors use a pushing algorithm to prove the generalised version of the shuffle theorem. An extension of that argument is used in [Iraci, Vanden Wyngaerd 2020] to formulate a valley version of the (generalised) Delta s…
We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of $Δ_{e_k} e_n$ at $q=1$ in terms of the elementary symmetric function basis. We then use a weight-preserving bijection to prove the Delta Conjecture at $q=1$. The method of proof provides a variety of structures which can compute the inner product of $Δ_{e_k} e_n|_{q=1}$ with any symmetric function.
By finding orthogonal representation for a family of simple connected called $δ$-graphs it is possible to show that $δ$-graphs satisfy delta conjecture. An extension of the argument to graphs of the form $\overline{P_{Δ(G)+2}\sqcup G}$ where $P_{Δ(G)+2}$ is a path and $G$ is a simple connected graph it is possible to find an orthogonal representation of $\overline{P_{Δ(G)+2}\sqcup G}$ in $\mathbb{R}^{Δ(G)+1}$. As a consequence we prove delta conjecture.
We conjecture a formula for the symmetric function $\frac{[n-k]_t}{[n]_t}Δ_{h_m}Δ_{e_{n-k}}ω(p_n)$ in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington (Loehr, Warrington 2007), recently proved by Sergel (Sergel 2017) after the breakthrough of Carlsson and Mellit (Carlsson, Mellit 2018). Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson (Haglund, Remmel, Wilson 2015), answering one of their questions. We support our conjecture by prov…
Using the celebrated Morris Constant Term Identity, we deduce a recent conjecture of Chan, Robbins, and Yuen (math.CO/9810154), that asserts that the volume of a certain $n(n-1)/2$-dimensional polytope is given by the product of the first n-1 Catalan numbers.
We give bounds on the L(2,1)-labeling number of a simple graph in terms of its order and its maximum degree. We also describe an infinite class of graphs of which the elements have the highest L(2,1)-labeling numbers in terms of their maximum degrees of any known infinite class of graphs.
In this paper we study the tangent spaces of the smooth nested Hilbert scheme $ Hil{n,n-1}$ of points in the plane, and give a general formula for computing the Euler characteristic of a $\TT^2$-equivariant locally free sheaf on $\Hil{n,n-1}$. Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables $q$ and $t$ with non-negative integer coefficients . We call this conjecturally positive polynomial as \textsl{the nested $q,t$-Cat alan series}, for it has many conjectural properties similar to that of the $q,t $-Catalan series.
We introduce the family of Theta operators $Θ_f$ indexed by symmetric functions $f$ that allow us to conjecture a compositional refinement of the Delta conjecture of Haglund, Remmel and Wilson for $Δ_{e_{n-k-1}}'e_n$. We show that the $4$-variable Catalan theorem of Zabrocki is precisely the Schröder case of our compositional Delta conjecture, and we show how to relate this conjecture to the Dyck path algebra introduced by Carlsson and Mellit, extending one of their results. Again using the Theta operators, we conjecture a touching refinement of the generalized Delta conjecture for $Δ_{h_m}Δ_{…
We prove the Schröder case, i.e. the case $\langle \cdot,e_{n-d}h_d \rangle$, of the conjecture of Haglund, Remmel and Wilson (Haglund et al. 2018) for $Δ_{h_m}Δ_{e_{n-k-1}}'e_n$ in terms of decorated partially labelled Dyck paths, which we call \emph{generalized Delta conjecture}. This result extends the Schröder case of the Delta conjecture proved in (D'Adderio, Vanden Wyngaerd 2017), which in turn generalized the $q,t$-Schröder of Haglund (Haglund 2004). The proof gives a recursion for these polynomials that extends the ones known for the aforementioned special cases. Also, we give another …
We discuss the combinatorics of the decorated Dyck paths appearing in the Delta conjecture framework of Haglund, Remmel and Wilson (2015) and Zabrocki (2016), by introducing two new statistics, bounce and bounce'. We then provide plethystic formulae for their q,t-enumerators, by proving in this way a decorated version of Haglund's q,t-Schroeder theorem, answering a question in Haglund, Remmel and Wilson (2015). In particular we provide both an algebraic and a combinatorial explanation of a symmetry conjectured in Haglund, Remmel and Wilson (2015) and Zabrocki (2016). Finally, we show a link be…
We will look at the Catalan numbers from the {\it Rigged Configurations} point of view originated \cite{Kir} from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models . Our strategy is to take a combinatorial interpretation of Catalan numbers $C_n$ as the number of standard Young tableaux of rectangular shape $(n^2)$, or equivalently, as the Kostka number $K_{(n^2),1^{2n}}$, as the starting point of research. We observe that the rectangular (or multidimensional) Catalan numbers $ C(m,n)$ introduced and studied by P. MacMahon \cit…
In this case-study in computer-human collaboration, we develop, implement, and execute symbolic-computational algorithms for the automatic discovery and proof of explicit expressions for the expectation, variance, and higher moments of a large class of natural combinatorial statistics defined on Catalan-counted objects, enabling, inter-alia, to prove that they are not asymptotically normal. In particular, we reproduce in 0.12 seconds results of Miklos Bona, and derive far deeper results, way beyond the scope of humans, concerning higher moments of the random variable "number of occurrences of …
In this note we use the analogy between the Catalan sequence and the Rueppel sequence to derive a variety of conjectures surrounding the Hankel transforms of a number of sequences closely related to the Rueppel sequence. Use is made of the representation of suitable generating functions by Stieltjes continued fractions. We define polynomial sequences by introducing parameters that define generalized Rueppel sequences, and we show that such polynomials have coefficient arrays that are Riordan arrays. Finally we conjecture the form of a product of Hankel transforms arising from the Rueppel seque…
Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an infinite family of counterexamples for the conjecture and showed that essentially these are the only counterexamples. This is also obtained independently by Chumutov and Vignes-Tourneret and they posed a problem: it would be interesting to know whether the partial duality polynomial and the related conjectures would make sence for general delta-matroids. In thi…
In \cite{HRW15}, Haglund, Remmel, Wilson state a conjecture which predicts a purely combinatorial way of obtaining the symmetric function $Δ_{e_k}e_n$. It is called the Delta Conjecture. It was recently proved in \cite{GHRY} that the Delta Conjecture is true when either $q=0$ or $t=0$. In this paper we complete a work initiated by Remmel whose initial aim was to explore the symmetric function $Δ_{s_ν} e_n$ by the same methods developed in \cite{GHRY}. Our first need here is a method for constructing a symmetric function that may be viewed as a "combinatorial side" for the symmetric function $Δ…
We discuss the combinatorics of decorated Dyck paths and decorated parallelogram polyominoes, extending to the decorated case the main results of both [Haglund 2004] and [Aval et al. 2014]. This settles in particular the cases $\langle\cdot,e_{n-d}h_d\rangle$ and $\langle\cdot,h_{n-d}h_d\rangle$ of the Delta conjecture of Haglund, Remmel and Wilson (2018). Along the way, we introduce some new statistics, formulate some new conjectures, prove some new identities of symmetric functions, and answer a few open problems in the literature (e.g. from [Haglund et al. 2018], [Zabrocki 2016], [Aval et a…
We introduce area, bounce and dinv statistics on decorated parallelogram polyominoes, and prove that some of their q,t-enumerators match $\langle Δ_{h_m} e_{n+1}, s_{k+1,1^{n-k}} \rangle$, extending in this way the work in (Aval et al. 2014). Also, we provide a bijective connection between decorated parallelogram polyominoes and decorated labelled Dyck paths, which allows us to prove the combinatorial interpretation of the coefficient $\langleΔ_{e_{m+n-k-1}}'e_{m+n}, h_m h_n \rangle$ predicted by the Delta conjecture in (Haglund et al. 2015). Finally, we define a statistic pmaj on partially la…
