# Questions tagged [multinomial-coefficients]

for questions about multinomial coefficients

9
questions

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### simple formula for a finite sum of multinomial numbers

Let $k,m$ and $r$ be positive integers.
Define
$$\Omega(k,m,r) = \binom k {m-2r}\binom {k-m+2r} r$$
and
$$\Omega(k,m) = \sum_{r=\max\{0,m-k\}}^{[\frac{m}{2}]}\Omega(k,m,r).$$
Question.
1. Is $...

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### How to do a multinomial theorem sum faster

For example we have this question :
Find the coefficient of $x^6$ in the following
$\frac{\left(x^{2}+x+2\right)^{9}}{20}$
So using multinomial Theorem which is this :
$\left(x_{1}+x_{2}+\cdots+x_{...

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### Coefficients of $(2+x+x^2)^n$ from trinomial coefficients

I would like to be able to express the coefficients of $(2+x+x^2)^n$ in terms of the trinomial coefficients studied by Euler, ${n \choose \ell}_2 = [x^\ell](1+x+x^2)^n$ where $[x^\ell]$ denotes the ...

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### Asymptotics of multinomial coefficients

Binomial coefficients have a well known asymptotics (https://en.wikipedia.org/wiki/Binomial_coefficient#Bounds_and_asymptotic_formulas) given by $$\binom nk\sim\binom{n}{\frac{n}{2}} e^{-d^2/(2n)} \...

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### Combinatorics of merging sequences from multinomial coefficients

If you have $m$ sequences $a_{11},\dots,a_{1n_1}$ through $a_{m1},\dots,a_{mn_m}$ each sorted in ascending order (assume there are no duplicates) then there is an unique way to merge them.
How many ...

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399 views

### coefficients in the expansion of multivariable expression

Consider the expansion of the following $N$ variable expression
$$ D_N(z_1,\ldots,z_N)=\prod_{1\leq j<k\leq N}\left(1-\frac{z_j}{z_k}\right)\left( 1-\frac{z_k}{z_j} \right) $$
For example, in the ...

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### Why this equality holds?

Let ${\bf S} = (S_1,...,S_d) \in \mathcal{L}(E)^d$. We recall that the norm of $\|{\bf S}\|$ is defined by
\begin{eqnarray*}
\|{\bf S}\|
&=&\sup\left\{\bigg(\displaystyle\sum_{k=1}^d\|S_kx\|^2\...

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### Inequality that involves alternating sum of multinomial coefficients

I've encountered the following problem that involves alternating sums of multinomial coefficients. Let $$f(k)=\sum_{i=0}^{n-k}(-1)^i\binom{n}{k,i,n-k-i}(k+i)^\alpha$$
where $\binom{n}{k,i,n-k-i}=\frac{...

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209 views

### How can i justify this multinomial coefficient identity? [closed]

$$\sum_{ k\ge 1 } \sum_{ (s_1,...,s_k) } \binom h{ s_1 ,..., s_k }=\binom{m-1}{h-1}$$
where the second summation is taken over all choices of the numbers
$${ s }_{ 1 },...,{ s }_{ k-1 }\ge 0,\quad {s ...