Sovardipk

2022-06-29

Monomials with degree k of the following polynomial $\left(1+{𝑥}_{1}+\cdots +{𝑥}_{1}^{q}+{y}_{1}+\cdots +{y}_{1}^{d}{\right)}^{n}$

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Dayana Zuniga

Expert

Move s and s′ as you wish as

using stars and bars(and inclusion-exclusion), we know that the number of tuples adding to s′ with parts less or equal than q is
$\sum _{r=0}^{s}\left(-1{\right)}^{r}\left(\genfrac{}{}{0}{}{s}{r}\right)\left(\genfrac{}{}{0}{}{{s}^{\prime }-\left(q+1\right)r+s-1}{s-1}\right),$
similarly for the ${y}_{j}$ notice that we are not allowing 0 as a part (I took $1={x}_{i}^{0}$) and we have
$\sum _{\ell =0}^{n-s}\left(-1{\right)}^{\ell }\left(\genfrac{}{}{0}{}{n-s}{\ell }\right)\left(\genfrac{}{}{0}{}{k-{s}^{\prime }-d\ell -1}{n-s-1}\right).$
Plugging all together, we get
$\sum _{s=0}^{n}\sum _{{s}^{\prime }=0}^{k}\sum _{r=0}^{s}\sum _{\ell =0}^{n-s}\left(-1{\right)}^{r+\ell }\left(\genfrac{}{}{0}{}{n}{s}\right)\left(\genfrac{}{}{0}{}{s}{r}\right)\left(\genfrac{}{}{0}{}{{s}^{\prime }-\left(q+1\right)r+s-1}{s-1}\right)\left(\genfrac{}{}{0}{}{n-s}{\ell }\right)\left(\genfrac{}{}{0}{}{k-{s}^{\prime }-d\ell -1}{n-s-1}\right)$
Not entirely sure if this sum simplifies.

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