MathDB

Let

n

be a positive integer. Suppose

S

is a set of ordered n-\mbox{tuples} of nonnegative integers such that, whenever

(a_1,\dots,an)\in S

and

b_i

are nonnegative integers with

b_i\le a_i

, the

n-\text{tuple}

(b_1,\dots,b_n)

is also in

S

. If

h_m

is the number of elements of

S

with the sum of components equal to

m

, prove that

h_m

is a polynomial in

m

for all sufficiently large

m

Problem Statement