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We prove an identity about partitions with a very elementary formulation. We had previously conjectured this identity, encountered in the study of shifted Jack polynomials (math.CO/9901040). The proof given is using a trivariate generating function. It would be interesting to obtain a bijective proof. We present a conjecture generalizing this identity.
Using techniques from integrable systems, we obtain a number of exact results for random partitions. In particular, we prove a simple formula for correlation functions of what we call the Schur measure on partitions (which is a far reaching generalization of the Plancherel measure, see math.CO/9905032) and also show that these correlations functions are tau-functions for the Toda lattice hierarchy. Also we give a new proof of the formula due to Bloch and the author, see alg-geom/9712009, for the so called n-point functions of the uniform measure on partitions and comment on the local structure of a typical partition.
We define a number of new combinatorial operations on skew semistandard domino tableaux, which together with constructions introduced earlier by C. Carre and B. Leclerc, define an elegant structure on the set of these tableaux, that closely parallels the structure defined in [math.CO/9908099] for ordinary skew semistandard tableaux, that is related to the Littlewood-Richardson rule. These operations are: (1) a bijection between semistandard domino tableaux and certain pairs of ordinary tableaux of the same weight that together fill the same shape; (2) a weight preserving transformation of domino tableaux to ordinary tableaux of a related shape (the correspondence involves 2-quotients) with the property that Yamanouchi domino tableaux correspond precisely to Littlewood-Richardson tableaux; (3) a correspondence between Yamanouchi domino tableaux of shape $λ$ and weight $μ$ and Yamanouchi domino tableaux of shape $μ'$ and weight $λ$, where $μ'$ is $μ$ scaled by a factor 2. The property at the heart of the structure so defined is that operations (1) and (2) commute with the coplactic operations (implicitly) defined by Carre and Leclerc. (1) elucidates their bijection analogous to that of Robinson (a.k.a. the Robinson-Schensted correspondence), while the operation of moving chains in domino tableaux underlying (2) is very similar to jeu de taquin.
The space $M_{μ/i,j}$ spanned by all partial derivatives of the lattice polynomial $Δ_{μ/i,j}(X;Y)$ is investigated in math.CO/9809126 and many conjectures are given. Here, we prove all these conjectures for the $Y$-free component $M_{μ/i,j}^0$ of $M_{μ/i,j}$. In particular, we give an explicit bases for $M_{μ/i,j}^0$ which allow us to prove directly the central {\sl four term recurrence} for these spaces.
Free Meixner states are a class of functionals on non-commutative polynomials introduced in math.CO/0410482. They are characterized by a resolvent-type form for the generating function of their orthogonal polynomials, by a recursion relation for those polynomials, or by a second-order non-commutative differential equation satisfied by their free cumulant functional. In this paper, we construct an operator model for free Meixner states. By combinatorial methods, we also derive an operator model for their free cumulant functionals. This, in turn, allows us to construct a number of examples. Many of these examples are shown to be trivial, in the sense of being free products of functionals which depend on only a single variable, or rotations of such free products. On the other hand, the multinomial distribution is a free Meixner state and is not a product. Neither is a large class of tracial free Meixner states which are analogous to the simple quadratic exponential families in statistics.
This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.
The set of weights of a finite-dimensional representation of a reductive Lie algebra has a natural poset structure ("weight poset"). Studying certain combinatorial problems related to antichains in weight posets, we realised that the best setting is provided by the representations associated with $\mathbb Z$-gradings of simple Lie algebras (arXiv: math.CO 1411.7683). If $\mathfrak g$ is a simple Lie algebra, then a $\mathbb Z$-grading of $\mathfrak g$ induces a $\mathbb Z$-grading of the corresponding root system $Δ$. In this article, we elaborate on a general theory of lower ideals (or antichains) in the corresponding weight posets $Δ(1)$. In particular, we provide a bijection between the lower ideals in $Δ(1)$ and certain elements of the Weyl group of $\mathfrak g$. An inspiring observation is that, to a great extent, the theory of lower ideals in $Δ(1)$ is similar to the theory of upper (= ad-nilpotent) ideals in the whole poset of positive roots $Δ^+$.
Jack polynomials in superspace, orthogonal with respect to a ``combinatorial'' scalar product, are constructed. They are shown to coincide with the Jack polynomials in superspace, orthogonal with respect to an ``analytical'' scalar product, introduced in hep-th/0209074 as eigenfunctions of a supersymmetric quantum mechanical many-body problem. The results of this article rely on generalizing (to include an extra parameter) the theory of classical symmetric functions in superspace developed recently in math.CO/0509408
For an affine algebra of nonexceptional type in the large rank we show the fermionic formula depends only on the attachment of the node 0 of the Dynkin diagram to the rest, and the fermionic formula of not type A can be expressed as a sum of that of type A with Littlewood-Richardson coefficients. Combining this result with math.CO/9901037 and arXiv:1002.3715 we settle the X=M conjecture under the large rank hypothesis.
We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudo-triangulation. This generalizes the main result of math.CO/0307347 which treats the minimally generically rigid case. The proof uses the concept of combinatorial pseudo-triangulation, CPT, in the plane and has two main steps: showing that a certain ``generalized Laman property'' is a necessary and sufficient condition for a CPT to be ``stretchable'', and showing that all generically rigid plane graphs admit a CPT assignment with that property. Additionally, we propose the study of combinatorial pseudo-triangulations on closed surfaces.
We describe the structure of 2-connected non-planar toroidal graphs with no K_{3,3}-subdivisions, using an appropriate substitution of planar networks into the edges of certain graphs called toroidal cores. The structural result is based on a refinement of the algorithmic results for graphs containing a fixed K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these graphs in linear-time and makes possible to enumerate labelled 2-connected toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex degree two or three by using an approach similar to [A. Gagarin, G. Labelle, and P. Leroux, "Counting labelled projective-planar graphs without a K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].
This paper, a continuation of math.CO/9909169, connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a Riemann-Hilbert problem and an associated system of integrable PDEs. In particular, we show that the Poissonization of the distribution function of this length can be identified as the Jimbo-Miwa-Ueno tau function.
Let $S\neq\mathbb N$ be a numerical semigroup generated by $e$ elements. In his paper (A Circle-Of-Lights Algorithm for the "Money-Changing Problem", Amer. Math. Monthly 85 (1978), 562--565), H.~S.~Wilf raised the following question: Let $Ω$ be the number of positive integers not contained in $S$ and $c-1$ the largest such element. Is it true that the fraction $\fracΩc$ of omitted numbers is at most $1-\frac1e$? Let $B\subseteq\mathbb N^{e-1}$ be the complement of an artinian $\mathbb N^{e-1}$-ideal. Following a concept of A.~Zhai (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1 [math.CO]) we relate Wilf's problem to a more general question about the weight distribution on $B$ with respect to a positive weight vector. An affirmative answer is given in special cases, similar to those considered by R.~Fröberg, C.~Gottlieb, R.~Häggkvist (On numerical semigroups, Semigroup Forum, Vol.~35, Issue 1, 1986/1987, 63--83) for Wilf's question.
Let $S$ be a numerical semigroup of embedding dimension $e$ and conductor $c$. The question of Wilf is, if $\#(\mathbb N\setminus S)/c\leq e-1/e$. \noindent In (An asymptotic result concerning a question of Wilf, arXiv:1111.2779v1 [math.CO], 2011, Lemma 3), Zhai has shown an analogous inequality for the distribution of weights $x\cdotγ$, $x\in\mathbb N^d$, w.\,r. to a positive weight vector $γ$: \noindent Let $B\subseteq\mathbb N^d$ be finite and the complement of an $\mathbb N^d$-ideal. Denote by $\operatorname{mean}(B\cdotγ)$ the average weight of $B$. Then \[\operatorname{mean}(B\cdotγ)/\max(B\cdotγ)\leq d/d+1.\] $\bullet$ For the family $Δ_n:=\{x\in\mathbb N^d|x\cdotγ<n+1\}$ of such sets we are able to show, that $\operatorname{mean}(Δ_n\cdotγ)/\max(Δ_n\cdotγ)$ converges to $d/d+1$, as $n$ goes to infinity. $\bullet$ Applying Zhai's Lemma 3 to the Hilbert function of a positively graded Artinian algebra yields a new class of numerical semigroups satisfying Wilf's inequality.
We study the mean-median map as a dynamical system on the space of finite sets of piecewise-affine continuous functions with rational coefficients. We determine the structure of the limit function in the neighbourhood of a distinctive family of rational points, the local minima. By constructing a simpler map which represents the dynamics in such neighbourhoods, we extend the results of Cellarosi and Munday (arXiv:1408.3454v1 [math.CO]) by two orders of magnitude. Based on these computations, we conjecture that the Hausdorff dimension of the graph of the limit function of the set $[0,x,1]$ is greater than 1.
We prove a central limit theorem for non-commutative random variables in a von Neumann algebra with a tracial state: Any non-commutative polynomial of averages of i.i.d. samples converges to a classical limit. The proof is based on a central limit theorem for ordered joint distributions together with a commutator estimate related to the Baker-Campbell-Hausdorff expansion. The result can be considered a generalization of Johansson's theorem on the limiting distribution of the shape of a random word in a fixed alphabet as its length goes to infinity [math.CO/9906120,math.PR/9909104].
For a Fano manifold M, complex conjugation defines a real involution on the quantum cohomology ring. For the Grassmannian we identify this involution with an explicit transformation on Schubert classes defined over the integers. It is a composition of a power of the cyclic action recently defined by Agnihotri and Woodward with Poincare duality. The existence of the involution has been observed independently and from a different point of view by Postnikov (math.CO/0205165).
We show that the distribution function of the first particle in a discrete orthogonal polynomial ensemble can be obtained through a certain recurrence procedure, if the (difference or q-) log-derivative of the weight function is rational. In a number of classical special cases the recurrence procedure is equivalent to the difference and q-Painleve equations of chao-dyn/9507010, [Sakai]. Our approach is based on the formalism of discrete integrable operators and discrete Riemann--Hilbert problems developed in math.CO/9912093, math-ph/0111008.