Let μ be a partition of k, and T a standard Young tableau of shape μ. McKay, Morse, and Wilf show that the probability a randomly chosen Young tableau of N cells contains T as a subtableau is asymptotic to fμ/k! as N goes to infinity, where fμ is the number of all tableaux of shape μ. We use a random-walk argument to show that the analogous asymptotic probability for randomly chosen Young table...

The widely studied q-polynomial fλ(q), which specializes when q = 1 to fλ, the number of standard Young tableaux of shape λ, has multiple combinatorial interpretations. It represents the dimension of the unipotent representation Sλ q of the finite general linear group GLn(q), it occurs as a special case of the Kostka-Foulkes polynomials, and it gives the generating function for the major index ...

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(∞) and of the Lie superalgebra osp(1|∞). A complete solution to the problem is presented, in which the Fock...

Combinatorial spiders are a model for the invariant space of the tensor product of representations. The basic objects, webs, are certain directed planar graphs with boundary; algebraic operations on representations correspond to graph-theoretic operations on webs. Kuperberg developed spiders for rank 2 Lie algebras and sl2. Building on a result of Kuperberg, Khovanov–Kuperberg found a recursive...

The Narayana numbers are N(n, k) = 1 n ( n k )( n k+1 ) . There are several natural statistics on Dyck paths with a distribution given by N(n, k). We show the equidistribution of Narayana statistics by computing the flag h-vector of J(2×n) in different ways. In the process we discover new Narayana statistics and provide co-statistics for which the Narayana statistics in question have a distribu...

This paper focuses on the combinatorial enumeration of truncated hollow standard Young tableaux approximate C shape. The formula approximately C-shaped SYT, namely convolution Catalan numbers, is derived by using multi-integral method based sequence statistical model. Meanwhile, it found that first and last rows differ in number boxes. In case where boxes row n large enough, SYT does not change...

This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S-matrix, Feynman diagrams, connected diagrams, Green functions, renormalization. The use of Hopf algebra for their definition allows for simple recursive derivations and lead ...