Search

Let s and t be variables. Define polynomials { n} in s, t by {0} = 0, {1} = 1, and { n} = s { n - 1} + t { n - 2} for n ≥ 2. If s, t are integers then the corresponding sequence of integers is called a Lucas sequence. Define an analogue of the binomial coefficients by where { n}! = {1} {2} ⋯ { n}. It is easy to see that is a polynomial in s and t. The purpose of this note is to give two combinatorial interpretations for this polynomial in terms of statistics on integer partitions inside a k × ( n - k) rectangle. When s = t = 1 we obtain combinatorial interpretations of the fibonomial coefficients which are simpler than any that have previously appeared in the literature. [ABSTRACT FROM AUTHOR]

The algebraic area probability distribution of closed planar random walks of length N on a square lattice is considered. The generating function for the distribution satisfies a recurrence relation in which the combinatorics is encoded. A particular case generalizes the q-binomial theorem to the case of three addends. The distribution fits the Lévy probability distribution for Brownian curves with its first-order 1/ N correction quite well, even for N rather small. [ABSTRACT FROM AUTHOR]

A set of new matrix Lie algebra and its corresponding loop algebra are constructed. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equation is generated. As its reduction cases, the multi-component Tu hierarchy is given. Finally, the multi-component integrable coupling system of Tu hierarchy is presented through enlarging matrix spectral problem. [ABSTRACT FROM AUTHOR]