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Any combinatorical meaning or interpretation of $$1^{\alpha_1}2^{\alpha_2}3^{\alpha_3}...s^{\alpha_s}\alpha_1!\alpha_2!...\alpha_s!$$ for partition $(1^{\alpha_1},2^{\alpha_2},3^{\alpha_3},...,s^{\alpha_s})\vdash{n}$.

In addition, this expression is diviser of $n!$.

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Given a permutation $(x_1,\dots,x_n)$ of numbers from 1 to $n$, we get a new permutation: $x_1$, $\dots$, $x_{\alpha_1}$ are fixed points, $(x_{\alpha_1+1},x_{\alpha_1+2})$ form a 2-cycle, and so on. Thus we get a permutation with $\alpha_i$ cycles of length $i$ and each of them is calculated as many times as you ask about.

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  • $\begingroup$ mmm, what about this $$\frac{n!}{1^{\alpha_1}2^{\alpha_2}3^{\alpha_3}...s^{\alpha_s}\alpha_1!\alpha_2!...\alpha_s!}$$ $\endgroup$
    – Radmir
    Commented Feb 28, 2015 at 15:06
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    $\begingroup$ It is a number of permutations with $\alpha_i$ cycles of length $i$. $\endgroup$
    – Fedor Petrov
    Commented Feb 28, 2015 at 15:11

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