I have a combinatorial question involving multivariate polynomials.
Let $n$ be a positive integer, and consider $n$ polynomials $p_1, p_2, \ldots, p_n$, where each $p_k$ is a dense polynomial in $k$ variables $x_1, \ldots, x_k$, consisting of all monomials of total degree $\leq d$. That is,
$$
p_k = \sum_{\substack{i_1 + \cdots + i_k \leq d}} x_1^{i_1} \cdots x_k^{i_k}.
$$
Let $T = p_1 p_2 \cdots p_n$ be the product of these polynomials. I am interested in counting how many distinct monomials appear in $T$ after multiplication.
Example
When $n = 2$ and $d = 2$, we have:
- $p_1 = 1 + x_1 + x_1^2$ (all degree $\leq 2$ monomials in $x_1$),
- $p_2 = 1 + x_1 + x_2 + x_1^2 + x_1 x_2 + x_2^2$ (all degree $\leq 2$ monomials in $x_1, x_2$)
Then the product $T = p_1 p_2$ is: $$ T = (1 + x_1 + x_1^2)(1 + x_1 + x_2 + x_1^2 + x_1 x_2 + x_2^2) $$
After expanding and combining like terms, the distinct monomials are:
$1$, $x_1$, $x_2$, $x_1^2$, $x_1 x_2$, $x_2^2$, $x_1^3$, $x_1^2 x_2$, $x_1 x_2^2$, $x_1^4$, $x_1^3 x_2$, $x_1^2 x_2^2$
So there are 12 distinct monomials.
Using SageMath, I computed the number of distinct monomials in $T$ for various $(n,d)$. Here are some results:
| $n$ | $d$ | $\#$ monomials in $T$ |
|---|---|---|
| 2 | 1 | 5 |
| 2 | 2 | 12 |
| 2 | 3 | 22 |
| 2 | 4 | 35 |
| 3 | 1 | 14 |
| 3 | 2 | 55 |
| 3 | 3 | 140 |
| 4 | 1 | 42 |
| 4 | 2 | 273 |
| 5 | 1 | 132 |
| ... | ... | ... |
Observation
Surprisingly, these numbers match the Fuss–Catalan numbers, which are defined by the formula: $$ C_n^{(d)} = \frac{1}{n(d - 1) + 1} \binom{nd}{n} $$
In my case, the number of monomials in $T = p_1 p_2 \cdots p_n$ appears to match the Fuss–Catalan number $C_{n+1}^{(d+1)}$:
$$
\#T = \frac{1}{(n+1)d + 1} \binom{(n+1)(d+1)}{n+1},
$$
with $n+1$ and $d+1$ as parameters.
For example:
- $n = 2$, $d = 2$ $\Rightarrow$ $C_3^{(3)} = \frac{1}{7} \binom{9}{3} = \frac{84}{7} = 12$.
Question
Why does the number of distinct monomials in the product $T = p_1 p_2 \cdots p_n$, with each $p_k$ being the complete polynomial of degree $\leq d$ in $x_1,\ldots,x_k$, match the $(n+1,d+1)$ Fuss–Catalan number?
I am aware of many combinatorial interpretations of the Fuss–Catalan numbers (e.g., $d$-ary trees, lattice paths, noncrossing partitions, etc.), but I haven't been able to relate any of them to this nested polynomial product structure.
Is there a bijective or algebraic explanation for this identity?
Any insight or reference would be greatly appreciated!