Enumeration of Standard Young Tableaux

A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.Comment: 65 pages, small correction

Set partitions, tableaux, and subspace profiles under regular split semisimple matrices

We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. At 1, they count set partitions with specified block sizes. At 0, they count standard tableaux of specified shape. At -1, they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express q-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions. For partitions whose parts are at most two, these polynomials are the non-zero entries of the Catalan triangle associated to the q-Hermite orthogonal polynomial sequence. In particular, when all parts are equal to two, they coincide with the polynomials defined by Touchard that enumerate chord diagrams by number of crossings.Comment: 28 pages, minor change

Generalized tableaux over arbitrary digraphs and their associated differential equations

We revisit the concepts of acyclic orderings and number of acyclic orderings of acyclic digraphs in terms of dispositions and counters for arbitrary multidigraphs. We prove that when we add a sequence of nested directed paths to a directed graph there is a unique polynomial such that the generatrix function of the family of counters is the product of the polynomial and the exponential function. We give an application, by considering a kind of digraphs arranged in rows introduced by the authors in a previous paper, called dispositional digraphs, in the particular case in which the digraph has two rows, to obtain new families of linear differential equations of small order whose coefficients are polynomials of small degree which admit polynomial solutions. In particular, we obtain a new differential equation associated to Catalan numbers, and the corresponding associated polynomials, which are solution of this differential equation; we term them Catalan differencial equation and Catalan polynomials, respectively. We prove that the Catalan polynomials obtained when we connect the directed path to the second vertex of the lower row of the digraph are orthogonal polynomials for an appropriate weight function. We characterize the digraphs that maximize the counter of connected dispositional digraphs and we find a new differential equation associated to these digraphs. We introduce also dispositions and counters in any multidigraph with non-strict inequalities in the dispositions, and we find new differential equations associated to some of them

Physical Combinatorics and Quasiparticles

We consider the physical combinatorics of critical lattice models and their associated conformal field theories arising in the continuum scaling limit. As examples, we consider A-type unitary minimal models and the level-1 sl(2) Wess-Zumino-Witten (WZW) model. The Hamiltonian of the WZW model is the $U_q(sl(2))$ invariant XXX spin chain. For simplicity, we consider these theories only in their vacuum sectors on the strip. Combinatorially, fermionic particles are introduced as certain features of RSOS paths. They are composites of dual-particles and exhibit the properties of quasiparticles. The particles and dual-particles are identified, through an energy preserving bijection, with patterns of zeros of the eigenvalues of the fused transfer matrices in their analyticity strips. The associated (m,n) systems arise as geometric packing constraints on the particles. The analyticity encoded in the patterns of zeros is the key to the analytic calculation of the excitation energies through the Thermodynamic Bethe Ansatz (TBA). As a by-product of our study, in the case of the WZW or XXX model, we find a relation between the location of the Bethe root strings and the location of the transfer matrix 2-strings.Comment: 57 pages, in version 2: typos corrected, some sentences clarified, one appendix remove

kk-Schur functions and affine Schubert calculus

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculusComment: 213 pages; conference website: http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/, updates and corrections since v1. This material is based upon work supported by the National Science Foundation under Grant No. DMS-065264

Combinatorial Structures in Random Matrix Theory Predictions for LL-Functions

Our results can be viewed as applications of algebraic combinatorics in random matrix theory. These applications are motivated by the predictive power of random matrix theory for the statistical behavior of the celebrated Riemann $\zeta$-function (and $L$-functions in general), which was discovered by Montgomery (with regard to zeros of $L$-functions) and by Keating and Snaith (with regard to values of $L$-functions). The first results revolve around a new operation on partitions, which we call overlap. We prove two overlap identities for so-called Littlewood-Schur functions. The first overlap identity represents the Littlewood-Schur function $LS _\lambda(X; Y)$ as a sum over subsets of $X$, while the second overlap identity essentially represents $LS_\lambda(X; Y)$ as a sum over pairs of partitions whose overlap equals $\lambda$. Both identities are derived by applying Laplace expansion to a determinantal formula for Littlewood-Schur functions due to Moens and Van der Jeugt. In addition, we give two visual characterizations for the set of all pairs of partitions whose overlap is equal to a partition $\lambda$. The second result is an asymptotic formula for averages of mixed ratios of characteristic polynomials over the unitary group, where mixed ratios are products of ratios and/or logarithmic derivatives. Our proof of this formula is a generalization of Bump and Gamburd's elegant combinatorial proof of Conrey, Forrester and Snaith's formula for averages of ratios of characteristic polynomials over the unitary group. The generalization relies on three combinatorial results, namely the first overlap identity, a new variant of the Murnaghan-Nakayama rule and an idea from vertex operator formalism. We conclude this thesis by explaining how this approach might lead to new number theoretic proofs.Comment: 155 pages, PhD thesi

T-systems and Y-systems in integrable systems

The T and Y-systems are ubiquitous structures in classical and quantum integrable systems. They are difference equations having a variety of aspects related to commuting transfer matrices in solvable lattice models, q-characters of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras with coefficients, periodicity conjectures of Zamolodchikov and others, dilogarithm identities in conformal field theory, difference analogue of L-operators in KP hierarchy, Stokes phenomena in 1d Schr\"odinger problem, AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace sequence in discrete geometry, Fermionic character formulas and combinatorial completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics, analytic and thermodynamic Bethe ans\"atze, quantum transfer matrix method and so forth. This review article is a collection of short reviews on these topics which can be read more or less independently.Comment: 156 pages. Minor corrections including the last paragraph of sec.3.5, eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical review) also needs these correction