Assumptions about the costs of character change, coded in the form of a step matrix, determine most-parsimonious inferences of character evolution on phylogenies. We present a graphical approach to exploring the relationship between cost assumptions and evolutionary inferences from character data. The number of gains and losses of a binary trait on a phylogeny can be plotted over a range of cos...

We provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible. It follows from our general bound that if a permutation σ ∈ Sn has at most n cycles of length < m, then |χ(σ)| ≤ χ(1) for all irreducible characters χ of Sn. This is a far reaching generalization of a result of Fomin and Lulov. We then use our various charact...

The characters of the irreducible representations of the symmetric group play an important role in many areas of mathematics. However, since the early work of Frobenius [5] in 1900, no explicit formula was found for them. The characters of the symmetric group were computed through various recursive algorithms, but explicit formulas were only known for about ten particular cases [5, 9]. The purp...

Billey, Jockusch, and Stanley characterized 321-avoiding permutations by a property of their reduced decompositions. This paper generalizes that result with a detailed study of permutations via their reduced decompositions and the notion of pattern containment. These techniques are used to prove a new characterization of vexillary permutations in terms of their principal dual order ideals in a ...

Motivated by the problem of constructing bijective maps with low differential uniformity, we introduce notion permutation resemblance a function, which looks to measure distance given map is from being permutation. We prove several results concerning permutation resemblance and show how it can ...