MathDB
Poset on subsets of 2n element set

Source: 2019 USAMO Problem 4/USAJMO Problem 5

April 18, 2019
AMCUSA(J)MOUSAMOOlympiad Combinatoricsrecursion2019 USAJMO2019 USAMO

Problem Statement

Let nn be a nonnegative integer. Determine the number of ways that one can choose (n+1)2(n+1)^2 sets Si,j⊆{1,2,…,2n}S_{i,j}\subseteq\{1,2,\ldots,2n\}, for integers i,ji,j with 0≤i,j≤n0\leq i,j\leq n, such that:
[*] for all 0≤i,j≤n0\leq i,j\leq n, the set Si,jS_{i,j} has i+ji+j elements; and [*] Si,j⊆Sk,lS_{i,j}\subseteq S_{k,l} whenever 0≤i≤k≤n0\leq i\leq k\leq n and 0≤j≤l≤n0\leq j\leq l\leq n.
Proposed by Ricky Liu
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