Up-Down Sequences of Permutations for Gene Expressions

Abstract

We will describe modifications of the
research methods of Willbrand et al and Ahnert et
al for identifying significant genes in the biological
processes studied in microarray experiments.
Willbrand et al introduced a new method of
identifying significant genes by analyzing
probabilities of up-down signatures of microarray
expression curves of genes. Ahnert et al generalized
the method of Willbrand et al and established
various bounds on any microarray curve’s
algorithmic compressibility which measures its
significance in underlying biological process. We
will compute the probabilities of up-down
signatures of microarray curves defined by
Willbrand et al by using Foulkes’ method for
enumeration of permutations with prescribed updown
sequences and the hook length formula of
Frame et al. Moreover, we will compute the bound
of Ahnert et al corresponding to the map which
gives the number of permutations with the same
pattern of rises and falls for any microarray
curve’s algorithmic compressibility. It is
fascinating to see that how combinatorial
algorithms of permutations and Young tableaux
are useful in analyzing data of gene expressions
and identifying significant genes in biological
processes.