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4
questions
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A summation formula for number of ways $n$ identical objects can be put in $m$ identical bins
A famous counting problem is to calculate the number of ways $n$ identical objects can be put into $m$ identical bins. I know that this problem is somewhat equivalent to Partition problem. There is no ...
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1
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summation of product of binomials coefficients over compositions
I am having trouble with this problem which arises in the context of computing lowest theoretically possible computation cost for some cryptographic primitive.
Let $n$ and $a$ be positive integers ...
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3
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Reorganising alternating sum of products of binomial coefficients
The summation
$$
\sum_{k=0}^{\lfloor\frac{p}{s}\rfloor}(-1)^k {n\choose k}{p-ks+n-1\choose n-1}\quad;n > 0, s > 0 \text{ and } 0\le p\le ns,
$$
with ${n\choose k}$ denoting the binomial ...
2
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Simplified $\sum_{n=a_1k_1+a_2k_2+\cdots+a_mk_m}\frac{(a_1+a_1+\cdots+a_m)!}{a_1!\cdot a_2!\cdots a_m!}\prod_{i=1}^m \left(f(k_i)\right)^{a_i}$
I'm studying a function of the form
$$b_n=\sum_{n=a_1k_1+a_2k_2+\cdots+a_mk_m}\frac{(a_1+a_2+\cdots+a_m)!}{a_1!\cdot a_2!\cdots a_m!}\prod_{i=1}^m \left(f(k_i)\right)^{a_i}$$
Where the sum is over ...
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