If the integer n is partitioned into a sum in which "1" appears j1 times, "2" appears j2 times, and so on, then the number of partitions of a set of size n that collapse to that partition of the integer n when the members of the set become indistinguisbable is the corresponding coefficient in the polynomial.
Examples
For example, we have
because there are
6 ways to partition of set of 6 as 5+1,
15 ways to partition of set of 6 as 4+2, and
10 ways to partition a set of 6 as 3+3.
Similarly,
because there are
15 ways to partition a set of 6 as 4+1+1,
60 ways to partition a set of 6 as 3+2+1, and
15 ways to partition a set of 6 as 2+2+2.
Bell numbers
The sum
is the nth Bell number, which is the number of partitions of a set of size n.
Relation to Faà di Bruno's formula
The coefficients in these polynomials are the Faà di Bruno coefficients, occurring in Faà di Bruno's formula for the nth derivative of a composition of two functions.
In the Western world, its most classical form is a church bell or town bell, which is hung within a tower and sounded by having the entire bell swung by ropes, whereupon an internal hinged tongue strikes the body of the bell (called a free-swinging bell).
In the Eastern world, the traditional forms of bells are temple and palace bells, small ones being rung by a sharp rap with a stick, and very large ones rung by a blow from the outside by a large swinging beam.
Sigismund is a bell in the Wawel Cathedral in Kraków, Poland, cast in 1520.
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