Let W be a Weyl group with root lattice Q and Coxeter number h. The elements of the finite torus Q/(h+1)Q are called the W -parking functions, and we call the permutation representation of W on the set of W -parking functions the (standard) W -parking space. Parking spaces have interesting connections to enumerative combinatorics, diagonal harmonics, and rational Cherednik algebras. In this pap...

The nth Delannoy number and the nth Schröder number given by D n = n k=0 n k n + k k and S n = n k=0 n k n + k k 1 k + 1 respectively arise naturally from enumerative combinatorics. Let p be an odd prime. We mainly show that p−1 k=1 D k k 2 ≡ 2 −1 p E p−3 (mod p) and p−1 k=1 S k m k ≡ m 2 − 6m + 1 2m 1 − m 2 − 6m + 1 p (mod p), where (−) is the Legendre symbol, E 0 , E 1 , E 2 ,. .. are Euler n...

Abstract We study the asymptotic behaviour of random factorizations n -cycle into transpositions fixed genus $$g>0$$ g > 0 . They have a geometric interpretation as branched covers sphere and their enumeration Hurwitz numbers was extensively studi...

Article history: Received 11 December 2006 Available online 16 April 2009

We construct a family of weight functions on finite abelian groups that yield invertible MacWilliams identities for additive codes. The weights are obtained composing a suitable support map with the rank function of a graded lattice that satisfies certain regularity properties. We express the Krawtchouk coefficients of the corresponding MacWilliams transformation in terms of the combinatorial i...

Just after almost simultaneous publications in 1979 of two books, " Lattice Path Combinatorics with Statistical Applications " , by T.V. Narayana and " Lattice Path Counting and Applications " , by me, I realized that there was a substantial growing interest in lattice path combinatorics and applications in the fields of applied probability, statistics and computer science. I also realized that...

we give a survey of recent applications of group rings to combinatorics and tocryptography, including their use in the di erential cryptanalysis of block ciphers.