is the number of ways of placing distinct objects into labeled boxes so that there are objects in the th box. It is also the number of -dimensional lattice paths from to . For , the multinomial coefficient is defined to be 1. For
and in general,
Table 26.4.1 gives numerical values of multinomials and partitions for . These are given by the following equations in which are nonnegative integers such that
is a partition of :
is the multinominal coefficient (26.4.2):
is the number of permutations of with cycles of length 1, cycles of length 2, , and cycles of length :
(The empty set is considered to have one permutation consisting of no cycles.) is the number of set partitions of with subsets of size 1, subsets of size 2, , and subsets of size :
For each all possible values of are covered.
where the summation is over all nonnegative integers such that .