Questions tagged [multinomial-coefficients]

for questions about multinomial coefficients

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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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1answer
205 views

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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3answers
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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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0answers
103 views

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 ...
8
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2answers
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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297 views

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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1answer
184 views

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 ...