Young's partition lattice $L(m,n)$ consists of unordered partitions having $m$ parts where each part is at most $n$. Using methods from complex algebraic geometry, R. Stanley proved that $L(m,n)$ is rank-symmetric, unimodal, and strongly Sperner. Moreover, he conjectured that $L(m,n)$ has a stronger property called symmetric chain decomposition. Despite many efforts, this conjecture has only been proved for $\min(m,n)\leq 4$. In this paper, we decompose $L(m,n)$ into level sets for certain tropical polynomials derived from the secant varieties of the rational normal curve in projective space, and we find that the resulting subposets have an elementary raising and lowering algorithm. As a corollary, we obtain a symmetric chain decomposition for the subposet of $L(m,n)$ consisting of "sufficiently generic" partitions.
We present a new characterization of the shape equivalent class and the Littlewood-Richardson correspondence of Young tableaux in terms of tropical (ultradiscrete) integrable systems. As an application, an alternative proof of the "shape change theorem" is given.
The map which takes a square matrix $A$ to its polytrope is piecewise linear. We show that cones of linearity of this map form a polytopal fan partition of $\{R}^{n \times n}$, whose face lattice is anti-isomorphic to the lattice of complete set of connected relations. This fan refines the non-fan partition of $\R^{n \times n}$ corresponding to cones of linearity of the eigenvector map. Our results answer open questions in a previous work with Sturmfels and lead to a new combinatorial classification of polytropes and tropical eigenspaces.
We show that the subgraph induced in Young's graph by the set of partitions with an odd number of standard Young tableaux is a binary tree. This tree exhibits self-similarities at all scales, and has a simple recursive description.
Simultaneous bar-cores, core shifted Young diagrams (or CSYDs), and doubled distinct cores have been studied since Morris and Yaseen introduced the concept of bar-cores. In this paper, our goal is to give a formula for the number of these core partitions on $(s,t)$-cores and $(s,s+d,s+2d)$-cores for the remaining cases that are not covered yet. In order to achieve this goal, we observe a characterization of $\bar{s}$-core partitions to obtain characterizations of doubled distinct $s$-core partitions and $s$-CSYDs. By using them, we construct $NE$ lattice path interpretations of these core part…
In this article we consider hook removal operators on odd partitions, i.e., partitions labelling odd-degree irreducible characters of finite symmetric groups. In particular we complete the discussion, started by Isaacs, Navarro, Olsson and Tiep in 2016, concerning the commutativity of such operators.
We compute the limit shapes of the Young diagrams of the minimal difference $p$ partitions and provide a simple physical interpretation for the limit shapes. We also calculate the asymptotic distribution of the largest part of the Young diagram and show that the scaled distribution has a Gumbel form for all $p$. This Gumbel statistics for the largest part remains unchanged even for general partitions of the form $E=\sum_i n_i i^{1/ν}$ with $ν>0$ where $n_i$ is the number of times the part $i$ appears.
An integer partition of $n$ is a decreasing sequence of positive integers that add up to $[n]$. Back in $1979$ Macdonald posed a question about the limit value of the probability that two partitions chosen uniformly at random, and independently of each other, are comparable in terms of the dominance order. In $1982$ Wilf conjectured that the uniformly random partition is a size-ordered degree sequence of a simple graph with the limit probability $0$. In $1997$ we showed that in both, seemingly unrelated, cases the limit probabilities are indeed zero, but our method left open the problem of con…
We extend the Andrews-Olsson identity to two-colored partitions. Regarding the sets of proper Young walls of quantum affine algebras $\g_n=A^{(2)}_{2n}$, $A^{(2)}_{2n-1}$, $B^{(1)}_{n}$, $D^{(1)}_{n}$ and $D^{(2)}_{n+1}$ as the sets of two-colored partitions, the extended Andrews-Olsson identity implies that the generating functions of the sets of reduced Young walls have very simple formulae: \begin{center} $\prod^{\infty}_{i=1}(1+t^i)^{κ_i}$ where $κ_i=0$, $1$ or $2$, and $κ_i$ varies periodically. \end{center} Moreover, we generalize the Bessenrodt's algorithms to prove the extended Andrews…
Let $p(n)$ be the number of all integer partitions of the positive integer $n$ and let $λ$ be a partition, selected uniformly at random from among all such $p(n)$ partitions. It is known that each partition $λ$ has a unique graphical representation, composed by $n$ non-overlapping cells in the plane called Young diagram. As a second step of our sampling experiment, we select a cell $c$ uniformly at random from among all $n$ cells of the Young diagram of the partition $λ$. For large $n$, we study the asymptotic behavior of the hook length $Z_n=Z_n(λ,c)$ of the cell $c$ of a random partituion $λ…
(Dual-)promotion and (dual-)evacuation are bijections on SYT(λ) for any partition λ. Let c^r denote the rectangular partition (c,...,c) of height r, and let sc_k (k > 2) denote the staircase partition (k,k-1,...,1). B. Rhoades showed representation-theoretically that promotion on SYT(c^r) exhibits the cyclic sieving phenomenon (CSP). In this paper, we demonstrate a promotion- and evacuation-preserving embedding of SYT(sc_k) into SYT(k^{k+1}). This arose from an attempt to demonstrate the CSP of promotion action on SYT(sc_k).
We study the restrictions, the strict fixed points, and the strict quotients of the partition complex $|Π_n|$, which is the $Σ_n$-space attached to the poset of proper nontrivial partitions of the set $\{1,\ldots,n\}$. We express the space of fixed points $|Π_n|^G$ in terms of subgroup posets for general $G\subset Σ_n$ and prove a formula for the restriction of $|Π_n|$ to Young subgroups $Σ_{n_1}\times \dots\times Σ_{n_k}$. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions. We uncover su…
In this paper, we study a family of generating functions whose coefficients are polynomials that enumerate partitions in lower order ideals of Young's lattice. Our main result is that this family satisfies a rational recursion and are therefore rational functions. As an application, we calculate the asymptotic behavior of the cardinality of lower order ideals for the ``average" partition of fixed length and give a homological interpretation of this result in relation to Grassmannians and their Schubert varieties.
Let $S_n$ be the symmetric group, and let $Y$ be a Young subgroup of $S_n$. Let $Π_n$ be the complex of partitions of $\{1, \ldots, n\}$. Our main result is a $Y$-equivariant decomposition of $Π_n$. As an application, we obtain new information about the quotient space of $|Π_n|$ by a Young subgroup.
A cascade of dihedral symmetries is hidden in Young's lattice of integer partitions. In fact, for each integer N>2 the Hasse graph of the subposet consisting of the partitions with maximal hook length strictly less than N has the dihedral group of order 2N as its symmetry group. Here a new interpretation of those Hasse graphs is presented, namely as the 1-skeleta of the injective hulls of certain finite metric spaces.
Using equivariant cohomology theory, Naruse obtained a hook length formula for the number of standard Young tableaux of skew shape $λ/μ$. Morales, Pak and Panova found two $q$-analogues of Naruse's formula respectively by counting semistandard Young tableaux of shape $λ/μ$ and reverse plane partitions of shape $λ/μ$. When $λ$ and $μ$ are both staircase shape partitions, Morales, Pak and Panova conjectured that the generating function of reverse plane partitions of shape $λ/ μ$ can be expressed as a determinant whose entries are related to $q$-analogues of the Euler numbers. The objective of th…
The conjugacy classes of nilpotent $n\times n$ matrices can be parametrised by partitions $λ$ of $n$, and for a nilpotent $η$ in the class parametrised by $λ$, the variety $F_η$ of $η$-stable flags has its irreducible components parametrised by the standard Young tableaux of shape $λ$. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Steinberg's result that the relative position of flags generically chosen in the irreducible components of $F_η$ parametrised by tableaux $P$ and $Q$, is the permutation associated to $(…
We introduce the notions of Schröder shape and of Schröder tableau, which provide some kind of analogs of the classical notions of Young shape and Young tableau. We investigate some properties of the partial order given by containment of Schröder shapes. Then we propose an algorithm which is the natural analog of the well known RS correspondence for Young tableaux, and we characterize those permutations whose insertion tableaux have some special shapes. The last part of the article relates the notion of Schröder tableau with those of interval order and of weak containment (and strong avoidance…
Our main result is a limit shape theorem for the two-dimensional surface defined by a uniform random n-by-n square Young tableau. The analysis leads to a calculus of variations minimization problem that resembles the minimization problems studied by Logan-Shepp, Vershik-Kerov, and Cohn-Larsen-Propp. Our solution involves methods from the theory of singular integral equations, and sheds light on the somewhat mysterious derivations in these works. An extension to rectangular diagrams, using the same ideas but involving some nontrivial computations, is also given. We give several applications of …
We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex operator methods of Okounkov--Reshetikhin--Vafa for the easy case G = Z/n; to handle the considerably more difficult case G=Z/2 x Z/2, we will also use a refinement of the author's recent q--enumeration of pyramid partitions. In the appendix, we relate the diagram generating functions to the Donaldson-Thomas partition functions of the orbifold C^3/G. We find …
We consider a new kind of straight and shifted plane partitions/Young tableaux --- ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of …
For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of partitions of $\ell$ with height $\leq k$, which is also the number of $(k+1)k\cdots21$-avoiding permutations of $\{1, 2, \ldots, \ell\}$. Finally, we apply this result to the representation theory of the affine Lie algebra $\widehat{sl}(n)$ and show that this quantity gives the multiplicity of certain maximal domin…
The z-measures on partitions originated from the problem of harmonic analysis of linear representations of the infinite symmetric group in the works of Kerov, Olshanski and Vershik (1993, 2004). A similar family corresponding to projective representations was introduced by Borodin (1997). The latter measures live on strict partitions (i.e., partitions with distinct parts), and the z-measures are supported by all partitions. In this note we describe some combinatorial relations between these two families of measures using the well-known doubling of shifted Young diagrams.
In this paper, we study shifted Schur functions $S_μ^\star$, as well as a new family of shifted symmetric functions $\mathfrak{K}_μ$ linked to Kostka numbers. We prove that both are polynomials in multi-rectangular coordinates, with nonnegative coefficients when written in terms of falling factorials. We then propose a conjectural generalization to the Jack setting. This conjecture is a lifting of Knop and Sahi's positivity result for usual Jack polynomials and resembles recent conjectures of Lassalle. We prove our conjecture for one-part partitions.
Plane partitions naturally appear in many problems of statistical physics and quantum field theory, for instance, in the theory of faceted crystals and of topological strings on Calabi-Yau threefolds. In this paper a connection is made between the exactly solvable model with the boson dynamical variables and a problem of enumeration of boxed plane partitions - three dimensional Young diagrams placed into a box of a finite size. The correlation functions of the boson model may be considered as the generating functionals of the Young diagrams with the fixed heights of its certain columns. The ev…
Lecture notes for the proceedings of the workshop "Algebraic Combinatorics related to Young diagram and statistical physics", Aug. 6-10 2012, I.I.A.S., Nara, Japan.
We study edge asymptotics of poissonized Plancherel-type measures on skew Young diagrams (integer partitions). These measures can be seen as generalizations of those studied by Baik--Deift--Johansson and Baik--Rains in resolving Ulam's problem on longest increasing subsequences of random permutations and the last passage percolation (corner growth) discrete versions thereof. Moreover they interpolate between said measures and the uniform measure on partitions. In the new KPZ-like 1/3 exponent edge scaling limit with logarithmic corrections, we find new probability distributions generalizing th…
The k-Young lattice Y^k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k>0. The Y^k poset was introduced in connection with generalized Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine symmetric group by a maximal parabolic subgroup. We prove a number of properties for $Y^k$ including that the covering relation is preserved when elements are translated by rectangular partitions with hook-length $k$. We highlight the order ideal generated by an $m\times n$ rectangular shape. This order ideal…
The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis ind…
We show that the space of algebraic covariant derivative curvature tensors R' is generated by Young symmetrized tensor products W*U or U*W, where W and U are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or {(1 1),(2 1)}. Each of the partitions (2), (3) and (1 1) describes exactly one symmetry class, whereas the partition (2 1) characterizes an infinite set S of irreducible symmetry classes. This set S contains exactly one symmetry class S_0 whose elements U can not play the role of gener…
The notion of a barely set-valued semistandard Young tableau was introduced by Reiner, Tenner and Yong in their study of the probability distribution of edges in the Young lattice of partitions. Given a partition $λ$ and a positive integer $k$, let ${\mathrm{BSSYT}}(λ,k)$ (respectively, ${\mathrm{SYT}}(λ,k)$) denote the set of barely set-valued semistandard Young tableaux (respectively, ordinary semistandard Young tableaux) of shape $λ$ with entries in row $i$ not exceeding $k+i$. In the case when $λ$ is a rectangular staircase partition $δ_d(b^a)$, Reiner, Tenner and Yong conjectured that $|{…
The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_λ^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young l…
Recently, Schneider and Schneider defined a new class of partitions called sequentially congruent partitions, in which each part is congruent to the next part modulo its index, and they proved two partition bijections involving these partitions. We introduce a new partition notation specific to sequentially congruent partitions which allows us to more easily study these bijections and their compositions, and we reinterpret them in terms of Young diagram transformations. We also define a generalization of sequentially congruent partitions, and we provide several new partition bijections for the…
The study of permutation and partition statistics is a classical topic in enumerative combinatorics. The major index statistic on permutations was introduced a century ago by Percy MacMahon in his seminal works. In this extended abstract, we study the well-known generalization of the major index to standard Young tableaux. We present several new results. In one direction, we introduce and study two partial orders on the standard Young tableaux of a given partition shape, in analogy with the strong and weak Bruhat orders on permutations. The existence of such ranked poset structures allows us t…
The up-operators $u_i$ and down-operators $d_i$ (introduced as Schur operators by Fomin) act on partitions by adding/removing a box to/from the $i$th column if possible. It is well known that the $u_i$ alone satisfy the relations of the (local) plactic monoid, and the present authors recently showed that relations of degree at most 4 suffice to describe all relations between the up-operators. Here we characterize the algebra generated by the up- and down-operators together, showing that it can be presented using only quadratic relations.
The concept of $t$-difference operator for functions of partitions is introduced to prove a generalization of Stanley's theorem on polynomiality of Plancherel averages of symmetric functions related to contents and hook lengths. Our extension uses a generalization of the notion of Plancherel measure, based on walks in the Young lattice with steps given by the addition of $t$-hooks. It is well-known that the hook lengths of multiples of $t$ can be characterized by the Littlewood decomposition. Our study gives some further information on the contents and hook lengths of other congruence classes …
Young's lattice $L(m,n)$ consists of partitions having $m$ parts of size at most $n$, ordered by inclusion of the corresponding Ferrers diagrams. K. O'Hara gave the first constructive proof of the unimodality of the Gaussian polynomials by expressing the underlying ranked set of $L(m,n)$ as a disjoint union of products of centered rank-unimodal subsets. We construct a finer decomposition which is compatible with the partial order on Young's lattice, at the cost of replacing the cartesian product with a more general poset extension. As a corollary, we obtain an explicit chain decomposition whic…
We show that the distribution of the major index over the set of involutions in S_n that avoid the pattern 321 is given by the q-analogue of the n-th central binomial coefficient. The proof consists of a composition of three non-trivial bijections, one being the Robinson-Schensted correspondence, ultimately mapping those involutions with major index m into partitions of m whose Young diagram fits inside an n/2 by n/2 box. We also obtain a refinement that keeps track of the descent set, and we deduce an analogous result for the comajor index of 123-avoiding involutions.
In 2002 R. Suter identified a dihedral symmetry on certain order ideals in Young's lattice and gave a combinatorial action on the partitions in these order ideals. Viewing this result geometrically, the order ideals can be seen to be seen to be in bijection with the alcoves in a 2-fold dilation in the geometric realization of the affine symmetric group. By considering the m-fold dilation we observe a larger set of order ideals in the k-bounded partition lattice that was considered by L. Lapointe, A. Lascoux, and J. Morse in the study of k-Schur functions. We identify the order ideal and the cy…
Let $GL_M$ be general linear Lie group over the complex field. The irreducible rational representations of the group $GL_M$ are labeled by pairs of partitions $μ$ and $\tildeμ$ such that the total number of non-zero parts of $μ$ and $\tildeμ$ does not exceed $M$. Let $U$ be the representation of $GL_M$ corresponding to such a pair. Regard the direct product $GL_N\times GL_M$ as a subgroup of $GL_{N+M}$. Let $V$ be the irreducible rational representation of the group $GL_{N+M}$ corresponding to a pair of partitions $λ$ and $\tildeλ$. Consider the vector space $W=Hom_{G_M}(U,V)$. It comes with a…
Walks on Young's lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at $\varnothing$, end at a row shape, and only visit partitions of bounded height are in bijection with a new type of arc diagram -- open diagrams. Remarkably two subclasses of open diagrams are equinumerous with well known objects: standard Young tableaux of bounded height, and Baxter permutations. We give an explicit combinatorial bijection in the former case.
We define a statistic called the weight of oscillating tableaux. Oscillating tableaux, a generalization of standard Young tableaux, are certain walks in Young's lattice of partitions. The weight of an oscillating tableau is the sum of the sizes of all the partitions that it visits. We show that the average weight of all oscillating tableaux of shape lambda and length 2n plus the size of lambda has a surprisingly simple formula: it is a quadratic polynomial in the size of lambda and n. Our proof via the theory of differential posets is largely computational. We suggest how the homomesy paradigm…
We consider the multivariate generating series $F_P$ of $P$-partitions in infinitely many variables $x_1, x_2 , \dots$. For some family of ranked posets $P$, it is natural to consider an analog $N_P$ with two infinite alphabets. When we collapse these two alphabets, we trivially recover $F_P$. Our main result is the converse, that is, the explicit construction of a map sending back $F_P$ onto $N_P$. We also give a noncommutative analog of the latter. An application is the construction of a basis of WQSym with a non-negative multiplication table, which lifts a basis of QSym introduced by K. Luo…
We study the number of $k \times r$ plane partitions, weighted on the sum of the first row. Using Erhart reciprocity, we prove an identity for the generating function. For the special case $k=1$ this result follows from the classical theory of partitions, and for $k=2$ it was proved in Andersson-Bhowmik with another method. We give an explicit formula in terms of Young tableaux, and study the corresponding zeta-function. We give an application on the average orders of towers of abelian groups. In particular we prove that the number of isomorphism classes of ``subgroups of subgroups of ... ($k-…
