Schröder Numbers: Paths, Partitions, and Domino Tilings

A tour of the large and little Schröder numbers: how they count lattice paths from (0,0) to (n,n) staying below the diagonal with steps (0,1),(1,0),(1,1); how they count guillotine partitions of a rectangle into n+1 pieces with n straight cuts; and the related Schröder paths with alternative steps. We’ll explain the simple relation S_n = 2 s_n for n > 0 between large and little Schröder numbers, and then dive into a stunning bridge to Aztec diamond tilings: the number of domino tilings of order n equals 2^{n(n+1)/2}, computable as the determinant of a Hankel matrix built from Schröder numbers. A single sequence weaving together lattice paths, partitions, and tilings across seemingly different combinatorial worlds.

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