  We define an enumerative function F(n,k,P,m) which is a generalization of
binomial coefficients. Special cases of this function are also power function,
factorials, rising factorials and falling factorials.
  The first section of the paper is an introduction.
  In the second section we derive an explicit formula for F. From the
expression for the power function we obtain a number theory result.
  Then we derive a formula which shows that the case of arbitrary m may be
reduced to the case m=0. This formula extends Vandermonde convolution.
  In the second section we describe F by the series of recurrence relations
with respect to each of arguments k, n, and P. As a special case of the first
recurrence relation we state a binomial identity. As a consequence of the
second recurrence relation we obtain relation for coefficients of Chebyshev
polynomial of both kind. This means that these polynomials might be defined in
pure combinatorial way.
