A generalized Goulden-Jackson cluster method and lattice path enumeration

The Goulden-Jackson cluster method for monoid networks and an application to lattice path enumeration

Given a nite or countably in nite set A, let A∗ be the set of all nite sequences of elements of A, including the empty sequence. We call A an alphabet, the elements of A letters, and the elements of A∗ words. By de ning an associative binary operation on two words by concatenating them, we see that A∗ is a monoid under the operation of concatenation, and we call A∗ the free monoid on A. The com...

Algebraic Methods for Permutations with Prescribed Patterns

Many problems which arise in a variety of disciplines may be expressed in terms of the enumeration of sequences, over a finite alphabet, which possess certain prescribed characteristics. Typical problems include the enumeration of non-self-intersecting paths on a rectangular lattice (a problem from crystal physics, Seymour and Welsh [ 16]), and the enumeration of sequences with no substrings in...

COMPUTATIONAL ENUMERATION OF POINT DEFECT CLUSTERS IN DOUBLE- LATTICE CRYSTALS

The cluster representation matrices have already been successfully used to enumerate close-packed vacancy clusters in all single-lattice crystals [I, 2]. Point defect clusters in double-lattice crystals may have identical geometry but are distinct due to unique atomic postions enclosing them. The method of representation matrices is extended to make it applicable to represent and enumerate ...

Reminiscing over

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...

The Application of Lagrangian Methods to the Enumeration of Labelled Trees with Respect to Edge Partition