Motivated by Stanley's results in \cite{St02}, we generalize the rank of a partition $λ$ to the rank of a shifted partition $S(λ)$. We show that the number of bars required in a minimal bar tableau of $S(λ)$ is max$(o, e + (\ell(λ) \mathrm{mod} 2))$, where $o$ and $e$ are the number of odd and even rows of $λ$. As a consequence we show that the irreducible projective characters of $S_n$ vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur's $Q_λ$ symmetric functions in terms of the power sum symmetric functions.
Bouvel and Pergola introduced the notion of minimal permutations in the study of the whole genome duplication-random loss model for genome rearrangements. Let $\mathcal{F}_d(n)$ denote the set of minimal permutations of length $n$ with $d$ descents, and let $f_d(n)= |\mathcal{F}_d(n)|$. They derived that $f_{n-2}(n)=2^{n}-(n-1)n-2$ and $f_n(2n)=C_n$, where $C_n$ is the $n$-th Catalan number. Mansour and Yan proved that $f_{n+1}(2n+1)=2^{n-2}nC_{n+1}$. In this paper, we consider the problem of counting minimal permutations in $\mathcal{F}_d(n)$ with a prescribed set of ascents. We show that suc…
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 $|{…
Baryshnikov and Romik derived the combinatorial identities for the numbers of the $m$-strip tableaux. This generalized the classical André's theorem for the number of up-down permutations. They asked for a bijective proof for the enumeration of $3$-strip tableaux. In this paper we will provide such a bijective proof. First we count the $3$-strip tableaux by decomposition. Secondly we will apply this "decomposition" idea on the up-down permutations and down-up permutations to enumerate the $3$-strip tableaux bijectively.
We give a bijective proof that the Betti numbers of a minimal free resolution of the Stanley-Reisner ring of a cycle graph (viewed as a one-dimensional simplicial complex) are given by the number of standard Young tableaux of a given shape.
We suggest an approach for the enumeration of minimal permutations having d descents which uses skew Young tableaux. We succeed in finding a general expression for the number of such permutations in terms of (several) sums of determinants. We then generalize the class of skew Young tableaux under consideration; this allows in particular to discover some presumably new results concerning Eulerian numbers.
We give a bijection between the symmetric group S_n, and the set of standard Young tableaux of rectangular shape m^n, m<=n, that have order n under jeu de taquin promotion.
In symmetric groups, a two-sided cell is the set of all permutations which are mapped by the Robinson-Schensted correspondence on a pair of tableaux of the same shape. In this article, we show that the set of permutations in a two-sided cell which have a minimal number of inversions is the set of permutations which have a maximal number of inversions in conjugated Young subgroups. We also give an interpretation of these sets with particular tableaux, called reading tableaux. As corollary, we give the set of elements in a two-sided cell which have a maximal number of inversions.
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 > 5; 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 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 thi…
We define a collection of functions $s_i$ on the set of plane trees (or standard Young tableaux). The functions are adapted from transpositions in the representation theory of the symmetric group and almost form a group action. They were motivated by $\textit{local moves}$ in combinatorial biology, which are maps that represent a certain unfolding and refolding of RNA strands. One main result of this study identifies a subset of local moves that we call $s_i$-local moves, and proves that $s_i$-local moves correspond to the maps $s_i$ acting on standard Young tableaux. We also prove that the gr…
The notion of an Abstract Young (briefly: AY) representation is a natural generalization of the classical Young orthogonal form. The AY representations of the symmetric group are characterized by Adin, Brenti and Roichman in [U2]. In this paper we present several types of minimal AY representation of $D_n$ associated with standard D-Young tableaux which are a natural generalization of usual standard Young tableaux. We give an explicit combinatorial view (the representation space is spanned by certain standard tableaux while the action is a generalized Young orthogonal form) of representations …
We investigate partitioning of integer sequences into heapable subsequences (previously defined and established by Mitzenmacher et al). We show that an extension of patience sorting computes the decomposition into a minimal number of heapable subsequences (MHS). We connect this parameter to an interactive particle system, a multiset extension of Hammersley's process, and investigate its expected value on a random permutation. In contrast with the (well studied) case of the longest increasing subsequence, we bring experimental evidence that the correct asymptotic scaling is $\frac{1+\sqrt{5}}{2…
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 …
The Schur functions in superspace $s_Λ$ and $\bar s_Λ$ are the limits $q=t=0$ and $q=t=\infty$ respectively of the Macdonald polynomials in superspace. We prove Pieri rules for the bases $s_Λ$ and $\bar s_Λ$ (which happen to be essentially dual). As a consequence, we derive the basic properties of these bases such as dualities, monomial expansions, and tableaux generating functions.
Nazarov and Tarasov recently generalized the notion of the rank of a partition to skew partitions. We give several characterizations of the rank of a skew partition and one possible characterization that remains open. One of the characterizations involves the decomposition of a skew shape into a minimal number of border strips, and we develop a theory of these MBSD's as well as of the closely related minimal border strip tableaux. An application is given to the value of a character of the symmetric group S_n indexed by a skew shape z at a permutation whose number of cycles is the rank of z.
Let $s_ν\circ s_μ$ denote the plethystic product of the Schur functions $s_ν$ and $s_μ$. In this article we define an explicit polynomial representation corresponding to $s_ν\circ s_μ$ with basis indexed by certain `plethystic' semistandard tableaux. Using these representations we prove generalizations of four results on plethysms due to Bruns--Conca--Varbaro, Brion, Ikenmeyer and the authors. In particular, we give a sufficient condition for the multiplicity $\langle s_ν\circ s_μ, s_λ\rangle$ to be stable under insertion of new parts into $μ$ and $λ$. We also characterize all maximal and mini…
We show that the shifted rank, or srank, of any partition $λ$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_λ$ function in terms of power sum symmetric functions. This gives an affirmative answer to a conjecture of Clifford. As pointed out by Clifford, the notion of the srank can be naturally extended to a skew partition $λ/μ$ as the minimum number of bars among the corresponding skew bar tableaux. While the srank conjecture is not valid for skew partitions, we give an algorithm to compute the srank.
A minimal code is a linear code where the only instance that a codeword has its support contained in the support of another codeword is when the codewords are scalar multiples of each other. Ashikhmin and Barg gave a sufficient condition for a code to be minimal, which led to much interest in constructing minimal codes that do not satisfy their condition. We consider a particular family of codes $\mathcal C_f$ when $f$ is the indicator function of a set of points, and prove a sufficient condition for $\mathcal C_f$ to be minimal and not satisfy Ashikhmin and Barg's condition based on certain g…
We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of $[n]$, as well as over all matchings on $[2n]$. As a corollary, the number of $k$-noncrossing partitions is equal to the number of $k$-nonnesting…
The minimal infinitesimal rigidity of bar-joint frameworks in the non-Euclidean spaces (R^2, ||.||_q) are characterised in terms of (2,2)-tight graphs. Specifically, a generically placed bar-joint framework (G,p) in the plane is minimally infinitesimally rigid with respect to a non-Euclidean l^q norm if and only if the underlying graph G = (V,E) contains 2|V|-2 edges and every subgraph H=(V(H),E(H)) contains at most 2|V(H)|-2 edges.
For a simplicial complex $Δ$ we study the effect of barycentric subdivision on ring theoretic invariants of its Stanley-Reisner ring. In particular, for Stanley-Reisner rings of barycentric subdivisions we verify a conjecture by Huneke and Herzog & Srinivasan, that relates the multiplicity of a standard graded $k$-algebra to the product of the maximal and minimal shifts in its minimal free resolution up to the height. On the way to proving the conjecture we develop new and list well known results on behavior of dimension, Hilbert series, multiplicity, local cohomology, depth and regularity whe…
