2,259 research outputs found
Combinatorially interpreting generalized Stirling numbers
Let be a word in alphabet with 's and 's.
Interpreting "" as multiplication by , and "" as differentiation with
respect to , the identity , valid
for any smooth function , defines a sequence , the terms of
which we refer to as the {\em Stirling numbers (of the second kind)} of .
The nomenclature comes from the fact that when , we have , the ordinary Stirling number of the second kind.
Explicit expressions for, and identities satisfied by, the have been
obtained by numerous authors, and combinatorial interpretations have been
presented. Here we provide a new combinatorial interpretation that retains the
spirit of the familiar interpretation of as a count of
partitions. Specifically, we associate to each a quasi-threshold graph
, and we show that enumerates partitions of the vertex set of
into classes that do not span an edge of . We also discuss some
relatives of, and consequences of, our interpretation, including -analogs
and bijections between families of labelled forests and sets of restricted
partitions.Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.00
A new upper bound on the game chromatic index of graphs
We study the two-player game where Maker and Breaker alternately color the
edges of a given graph with colors such that adjacent edges never get
the same color. Maker's goal is to play such that at the end of the game, all
edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored
edge where every color is blocked. The game chromatic index
denotes the smallest for which Maker has a winning strategy.
The trivial bounds hold for every
graph , where is the maximum degree of . In 2008, Beveridge,
Bohman, Frieze, and Pikhurko proved that for every there exists a
constant such that holds for any graph
with , and conjectured that the same
holds for every graph . In this paper, we show that is true for all graphs with . In
addition, we consider a biased version of the game where Breaker is allowed to
color edges per turn and give bounds on the number of colors needed for
Maker to win this biased game.Comment: 17 page
Set maps, umbral calculus, and the chromatic polynomial
Some important properties of the chromatic polynomial also hold for any
polynomial set map satisfying p_S(x+y)=\sum_{T\uplus U=S}p_T(x)p_U(y). Using
umbral calculus, we give a formula for the expansion of such a set map in terms
of any polynomial sequence of binomial type. This leads to some new expansions
of the chromatic polynomial. We also describe a set map generalization of Abel
polynomials.Comment: 20 page
- β¦