I encountered two Laurent polynomials
$$\sum_{j=0}^k\sum_{
\mbox{$\begin{array}{c}
i_1,\dots,i_k\geq 0\\
i_0=Ni_1\dotsi_k\end{array}$}}
\sum_{m=0}^{i_j}
\begin{pmatrix}
N\\
i_1,\dots,i_k
\end{pmatrix}
\begin{pmatrix}
i_j\\
m
\end{pmatrix}
x_0^{i_0}\cdots x^{i_jm}\cdots x_k^{i_k}
$$ and
$$
\sum_{j=0}^k\sum_{N\geq p_1\geq\cdots\geq p_{n}\geq 0}
\sum_{
\mbox{$\begin{array}{c}
i_1,\dots,i_k\geq 0\\
i_0=Ni_1\dotsi_kp_1\cdots p_{n}\end{array}$}}
\begin{pmatrix}
N\\
p_1
\end{pmatrix}
\begin{pmatrix}
p_1\\
p_2
\end{pmatrix}\cdots
\begin{pmatrix}
p_{n1}\\
p_{n}
\end{pmatrix}
\begin{pmatrix}
(N+p_1+\cdots+p_n)\\
i_1,\dots,i_k
\end{pmatrix}
x_0^{i_0}\cdots x^{i_j+p_1+\cdots+p_n}\cdots x_k^{i_k},
$$
where $N,n,k$ are given positive integers and
$$
\begin{pmatrix}
l\\
i_1,\dots,i_k
\end{pmatrix}=
\begin{pmatrix}
l\\
i_1
\end{pmatrix}
\begin{pmatrix}
li_1\\
i_2
\end{pmatrix}\cdots
\begin{pmatrix}
li_1\cdotsi_{k1}\\
i_k
\end{pmatrix}
$$
for any integer $l$, positive integers $i_1,\dots,i_k$.
I need to prove that they are actually the same.
Can anyone help me? or just give me some hint.
I'll be grateful.
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.


