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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.
35
votes
Accepted
How to prove this polynomial always has integer values at all integers?
$$P_m(x)=\sum_{i=0}^{m}\sum_{j=0}^{m}\binom{x+j}{ j}\binom{x-1}{ j}\binom{j}{ i}\binom{m}{ i}\binom{i}{ m-j}\frac{3}{(2i-1)(2j+1)(2m-2i-1)}.$$
Our task is to show it takes integer values on integers. …
53
votes
Conceptual reason why the sign of a permutation is well-defined?
Draw a braid on $n$ strings sloppily, so that no three strings pass through the same point. Convince yourself that the parity of the number of crossings is the same in every other drawing of the sam …