Enumeration via ballot numbers

Super Ballot Numbers

The Catalan numbers C n = (2n)!/n! (n + 1)! are are well-known integers that arise in many combinatorial problems. The numbers 6(2n)!/n! (n + 2)!, 60(2n)!/n! (n + 3)!, and more generally (2r + 1)!/r! · (2n)!/n! (n + r + 1)! are also integers for all n. We study the properties of these numbers and of some analogous numbers that generalize the ballot numbers, which we call super ballot numbers.

Labeled Ballot Paths and the Springer Numbers

The Springer numbers are defined in connection with the irreducible root systems of type Bn, which also arise as the generalized Euler and class numbers introduced by Shanks. Combinatorial interpretations of the Springer numbers have been found by Purtill in terms of André signed permutations, and by Arnol’d in terms of snakes of type Bn. We introduce the inversion code of a snake of type Bn an...

A curious polynomial interpolation of Carlitz-Riordan's q-ballot numbers

We study a polynomial sequence Cn(x|q) defined as a solution of a q-difference equation. This sequence, evaluated at q-integers, interpolates Carlitz–Riordan’s q-ballot numbers. In the basis given by some kind of q-binomial coefficients, the coefficients are again some qballot numbers. We obtain another curious recurrence relation for these polynomials in a combinatorial way.

Ballot Numbers, Alternating Products, and the Erdős-Heilbronn Conjecture

Frobenius Numbers by Lattice Point Enumeration

The Frobenius number g(A) of a set A = (a1, a2, . . . , an) of positive integers is the largest integer not representable as a nonnegative linear combination of the ai. We interpret the Frobenius number in terms of a discrete tiling of the integer lattice of dimension n−1 and obtain a fast algorithm for computing it. The algorithm appears to run in average time that is softly quadratic and we p...