Abstract A Richman game is a combinatorial game in which, rather than alternating moves, the two players bid for the privilege of making the next move. The theory of such games is a hybrid between… Expand

A Richman game is a combinatorial game in which, rather than alternating moves, the two players bid for the privilege of making the next move. We find optimal strategies for both the case where a… Expand

We will define an analog of a set which can contain either a positive or negative number of elements. We will allow sums to be calculated over an arbitrary hybrid set. This will lead us to a… Expand

We introduce the notion of a stable winning coalition in a multiplayer game as a new system of classification of games. An axiomatic refinement of this classification for three-player games is also… Expand

We survey the mathematical literature on umbral calculus (otherwise known as the calculus of finite differences) from its roots in the 19th century (and earlier) as a set of "magic rules" for… Expand

Abstract We find all sequences of polynomials (pn)n⩾0with persistent roots (i.e.,pn(x)=cn(x−r1)(x−r2)···(x−rn)) that are of binomial type in Viskov's generalization of Rota's umbral calculus to… Expand

The question of what is the best generalization of the factorial and the binomial coefficient is posed and several examples, derive their combinatorial properties, and demonstrate their interrelationships are given.Expand

The Stirling numbers of the first kind $s(a,k)$ are generalized to the case where $a$ may be an arbitrary real number, and new combinatorial properties held by the classical StirlingNumbers are discovered.Expand