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Questions tagged [multinomial-coefficients]

for questions about multinomial coefficients

8
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2answers
318 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 ...
0
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1answer
185 views

Norm of a tuple of operators

Let $F$ be a complex Hilbert space and $\mathcal{B}(F)$ be the algebra of all bounded linear operators on $F$. For ${\bf A} = (A_1,...,A_d) \in \mathcal{B}(F)^d$, the norm of ${\bf A}$ is given by $...
4
votes
1answer
288 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\...
0
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1answer
146 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{...
0
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1answer
183 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 ...