Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A matrix generalization of a theorem of Fine   In 1947 Nathan Fine gave a beautiful product for the number of binomial
coefficients $\binom{n}{m}$, for $m$ in the range $0 \leq m \leq n$, that are
not divisible by $p$. We give a matrix product that generalizes Fine's formula,
simultaneously counting binomial coefficients with $p$-adic valuation $\alpha$
for each $\alpha \geq 0$. For each $n$ this information is naturally encoded in
a polynomial generating function, and the sequence of these polynomials is
$p$-regular in the sense of Allouche and Shallit. We also give a further
generalization to multinomial coefficients.

21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
36
The categories of this paper are: cmp-lg
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
14
The categories of this paper are: math-ph
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom