Counting Staircases in Integer Compositions

Compositions of N with No Occurrence of K

A composition of n is an ordered collection of one or more positive integers whose sum is n. The number of summands is called the number of parts of the composition. A palindromic composition or palindrome is a composition in which the summands are the same in the given or in reverse order. Compositions may be viewed as tilings of 1-by-n rectangles with 1-by-i rectangles, 1 i n ≤ ≤ . We count t...

A Unified Approach to Algorithms Generating Unrestricted and Restricted Integer Compositions and Integer Partitions

An original algorithm is presented that generates both restricted integer compositions and restricted integer partitions that can be constrained simultaneously by a) upper and lower bounds on the number of summands (“parts”) allowed, and b) upper and lower bounds on the values of those parts. The algorithm can implement each constraint individually, or no constraints to generate unrestricted se...

Ham Sandwiches, Staircases and Counting Polynomials

Inside-Out Polytopes

We present a common generalization of counting lattice points in rational polytopes and the enumeration of proper graph colorings, nowhere-zero flows on graphs, magic squares and graphs, antimagic squares and graphs, compositions of an integer whose parts are partially distinct, and generalized latin squares. Our method is to generalize Ehrhart’s theory of lattice-point counting to a convex pol...

Forbidden substrings on weighted alphabets

In an influential 1981 paper, Guibas and Odlyzko constructed a generating function for the number of length n strings over a finite alphabet that avoid all members of a given set of forbidden substrings. Here we extend this result to the case in which the strings are weighted. This investigation was inspired by the problem of counting compositions of an integer n that avoid all compositions of ...