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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
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votes
Sums of binomial coefficients weighted by incomplete gamma
A combinatorial proof may not be easy, or even considered proper, but it could shed some light on the meaning of these formulae.
The sum with alternating signs resembles the definition of derangement …