Some relations of subsequences in permutations to graph theory with algorithmic applications

A thesis submitted to the Faculty d£ Science of the University of the Witwatersfand in fulfillment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 1977.The representation of some types of graphs as permutations, is utilized in devising efficient algorithms on those graphs. Maximum 'cliques in permutation graphs and circle graphs are found, by searching for a longest ascending or descending subsequence in their representing permutation. The correspondence between n-noded binary trees and the set SSn of stack-sortable permutations, forms the basis of an algorithm for generating and indexing such trees. The-relations between a graph and its representing p ermutation, are also employed in the proof of theorems concerning properties of subsequences in this permutation. In particular, expressions for the average lengths of the longest ascending and descending subsequence a in a random member of SSn , and the average number of inversions in such a permutation, are derived using properties of binary trees. Finally, a correspondence between the set SSn , and the set of permutations of order n With no descending subsequence of length 3, is demonstrated

Combinatorics of Oscillating Tableaux

In this paper, we first introduce the RSK algorithm, which gives a correspondence between integer sequences and standard tableaux. Then we introduce Schensted’s theorem and Greene’s theorem that describe how the shape of the standard tableau is determined by the sequence. We list four different bijections constructed by using the RSK insertion. The first one is a bijection between vacillating tableaux and pairs (P, T), where P is a set of ordered pairs and T is a standard tableau. The second one is a bijection between set partitions of [n] and vacillating tableaux. The third one is about partial matchings and up-down tableaux and the final one is from sequences to pairs (T, P), where T is still a standard tableau and P is a special oscillating tableau. We analyze some combinatorial statistics via these bijections

Decomposition of the product of a monomial and a Demazure atom

International audienceWe prove that the product of a monomial and a Demazure atom is a positive sum of Demazure atoms combinatorially. This result proves one particular case in a conjecture which provides an approach to a combinatorial proof of Schubert positivity property