In mathematics, the Schröder number S n , {\displaystyle S_{n},} also called a large Schröder number or big Schröder number, describes the number of lattice paths from the southwest corner ( 0 , 0 ) {\displaystyle (0,0)} of an n × n {\displaystyle n\times n} grid to the northeast corner ( n , n ) , {\displaystyle (n,n),} using only single steps north, ( 0 , 1 ) ; {\displaystyle (0,1);} northeast, ( 1 , 1 ) ; {\displaystyle (1,1);} or east, ( 1 , 0 ) , {\displaystyle (1,0),} that do not rise above the SW–NE diagonal. The first few Schröder numbers are
1, 2, 6, 22, 90, 394, 1806, 8558, ... (sequence A006318 in the OEIS). where S 0 = 1 {\displaystyle S_{0}=1} and S 1 = 2. {\displaystyle S_{1}=2.} They were named after the German mathematician Ernst Schröder.
Examples The following figure shows the 6 such paths through a 2 × 2 {\displaystyle 2\times 2} grid:
Related constructions A Schröder path of length n {\displaystyle n} is a lattice path from ( 0 , 0 ) {\displaystyle (0,0)} to ( 2 n , 0 ) {\displaystyle (2n,0)} with steps northeast, ( 1 , 1 ) ; {\displaystyle (1,1);} east, ( 2 , 0 ) ; {\displaystyle (2,0);} and southeast, ( 1 , − 1 ) , {\displaystyle (1,-1),} that do not go below the x {\displaystyle x} -axis. The n {\displaystyle n} th Schröder number is the number of Schröder paths of length n {\displaystyle n} . The following figure shows the 6 Schröder paths of length 2.
Similarly, the Schröder numbers count the number of ways to divide a rectangle into n + 1 {\displaystyle n+1} smaller rectangles using n {\displaystyle n} cuts through n {\displaystyle n} points given inside the rectangle in general position, each cut intersecting one of the points and dividing only a single rectangle in two (i.e., the number of structurally-different guillotine partitions). This is similar to the process of triangulation, in which a shape is divided into nonoverlapping triangles instead of rectangles. The following figure shows the 6 such dissections of a rectangle into 3 rectangles using two cuts:
Pictured below are the 22 dissections of a rectangle into 4 rectangles using three cuts:
The Schröder number S n {\displaystyle S_{n}} also counts the separable permutations of length n − 1. {\displaystyle n-1.}
Related sequences Schröder numbers are sometimes called large or big Schröder numbers because there is another Schröder sequence: the little Schröder numbers, also known as the Schröder-Hipparchus numbers or the super-Catalan numbers. The connections between these paths can be seen in a few ways:
Consider the paths from ( 0 , 0 ) {\displaystyle (0,0)} to ( n , n ) {\displaystyle (n,n)} with steps ( 1 , 1 ) , {\displaystyle (1,1),} ( 2 , 0 ) , {\displaystyle (2,0),} and ( 1 , − 1 ) {\displaystyle (1,-1)} that do not rise above the main diagonal. There are two types of paths: those that have movements along the main diagonal and those that do not. The (large) Schröder numbers count both types of paths, and the little Schröder numbers count only the paths that only touch the diagonal but have no movements along it. Just as there are (large) Schröder paths, a little Schröder path is a Schröder path that has no horizontal steps on the x {\displaystyle x} -axis. If S n {\displaystyle S_{n}} is the n {\displaystyle n} th Schröder number and s n {\displaystyle s_{n}} is the n {\displaystyle n} th little Schröder number, then S n = 2 s n {\displaystyle S_{n}=2s_{n}} for n > 0 {\displaystyle n>0} ( S 0 = s 0 = 1 ) . {\displaystyle (S_{0}=s_{0}=1).}
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