Zaskulnikov's identity

In mathematics, Zaskulnikov's identity (or sorting identity) is a relation between the ordered set of a set S of n numbers and the minima of the 2ⁿ − 1 nonempty subsets of S.

Let S = {x₁, x₂, ..., x_n} and $x_{1}<x_{2}<\dots <x_{n}$ .

The identity states that

\sum _{l=1}^{n}\lambda ^{l}x_{l}=\sum _{k=1}^{n}(1-\lambda )^{k-1}\lambda ^{n-k+1}\sum _{\mathrm {samp} }\min(x_{\alpha _{1}},x_{\alpha _{2}},\dots ,x_{\alpha _{k}})

where $0<\lambda <\infty$ and the inner sum is over all possible samples of $k$ elements of $n$ , or conversely

\sum _{l=1}^{n}\lambda ^{l}x_{l}=\sum _{k=1}^{n}(1-\lambda )^{k-1}\lambda ^{n-k+1}\sum _{\mathrm {samp} }\max(x_{\alpha _{1}},x_{\alpha _{2}},\dots ,x_{\alpha _{k}})

provided that $x_{1}>x_{2}>\dots >x_{n}$ .

Zaskulnikov's identity automatically arranges its left-hand side in ascending order of $x_{i}$ for the given right-hand side.

Zaskulnikov's identity generalizes the maximum-minimums identity reduces to it in the limit $\lambda \rightarrow \infty$ .

References

Zaskulnikov V. M., Statistical mechanics of fluids in a step potential: arXiv:1205.6546

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