Schröder number

In mathematics, the Schröder number ${\displaystyle S_{n},}$ also called a large Schröder number or big Schröder number, describes the number of lattice paths from the southwest corner ${\displaystyle (0,0)}$ of an ${\displaystyle n\times n}$ grid to the northeast corner ${\displaystyle (n,n),}$ using only single steps north, ${\displaystyle (0,1);}$ northeast, ${\displaystyle (1,1);}$ or east, ${\displaystyle (1,0),}$ that do not rise above the SW–NE diagonal.^[1]

Schröder number

Named after	Ernst Schröder
No. of known terms	infinity
First terms	1, 2, 6, 22, 90, 394, 1806
OEIS index	A006318 Large Schröder

The first few Schröder numbers are

1, 2, 6, 22, 90, 394, 1806, 8558, ... (sequence A006318 in the OEIS).

where ${\displaystyle S_{0}=1}$ and ${\displaystyle S_{1}=2.}$ They were named after the German mathematician Ernst Schröder.

Examples

The following figure shows the 6 such paths through a ${\displaystyle 2\times 2}$ grid:

Related constructions

A Schröder path of length ${\displaystyle n}$ is a lattice path from ${\displaystyle (0,0)}$ to ${\displaystyle (2n,0)}$ with steps northeast, ${\displaystyle (1,1);}$ east, ${\displaystyle (2,0);}$ and southeast, ${\displaystyle (1,-1),}$ that do not go below the ${\displaystyle x}$-axis. The ${\displaystyle n}$th Schröder number is the number of Schröder paths of length ${\displaystyle n}$.^[2] 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 ${\displaystyle n+1}$ smaller rectangles using ${\displaystyle n}$ cuts through ${\displaystyle n}$ 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 ${\displaystyle S_{n}}$ also counts the separable permutations of length ${\displaystyle n-1.}$

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 ${\displaystyle (0,0)}$ to ${\displaystyle (n,n)}$ with steps ${\displaystyle (1,1),}$ ${\displaystyle (2,0),}$ and ${\displaystyle (1,-1)}$ 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.^[3]
• Just as there are (large) Schröder paths, a little Schröder path is a Schröder path that has no horizontal steps on the ${\displaystyle x}$-axis.^[4]
• If ${\displaystyle S_{n}}$ is the ${\displaystyle n}$th Schröder number and ${\displaystyle s_{n}}$ is the ${\displaystyle n}$th little Schröder number, then ${\displaystyle S_{n}=2s_{n}}$ for ${\displaystyle n>0}$ ${\displaystyle (S_{0}=s_{0}=1).}$^[4]

Schröder paths are similar to Dyck paths but allow the horizontal step instead of just diagonal steps. Another similar path is the type of path that the Motzkin numbers count; the Motzkin paths allow the same diagonal paths but allow only a single horizontal step, (1,0), and count such paths from ${\displaystyle (0,0)}$ to ${\displaystyle (n,0)}$.^[5]

There is also a triangular array associated with the Schröder numbers that provides a recurrence relation^[6] (though not just with the Schröder numbers). The first few terms are

1, 1, 2, 1, 4, 6, 1, 6, 16, 22, .... (sequence A033877 in the OEIS).

It is easier to see the connection with the Schröder numbers when the sequence is in its triangular form:

k n	0	1	2	3	4	5	6
0	1
1	1	2
2	1	4	6
3	1	6	16	22
4	1	8	30	68	90
5	1	10	48	146	304	394
6	1	12	70	264	714	1412	1806

Then the Schröder numbers are the diagonal entries, i.e. ${\displaystyle S_{n}=T(n,n)}$ where ${\displaystyle T(n,k)}$ is the entry in row ${\displaystyle n}$ and column ${\displaystyle k}$. The recurrence relation given by this arrangement is

${\displaystyle T(n,k)=T(n,k-1)+T(n-1,k-1)+T(n-1,k)}$

with ${\displaystyle T(1,k)=1}$ and ${\displaystyle T(n,k)=0}$ for ${\displaystyle k>n}$.^[6] Another interesting observation to make is that the sum of the ${\displaystyle n}$th row is the ${\displaystyle (n+1)}$st little Schröder number; that is,

${\displaystyle \sum _{k=0}^{n}T(n,k)=s_{n+1}}$.

Recurrence relations

With ${\displaystyle S_{0}=1}$, ${\displaystyle S_{1}=2}$, ^[7]

${\displaystyle S_{n}=3S_{n-1}+\sum _{k=1}^{n-2}S_{k}S_{n-k-1}}$ for ${\displaystyle n\geq 2}$

and also ^[8]

${\displaystyle S_{n}={\frac {6n-3}{n+1}}S_{n-1}-{\frac {n-2}{n+1}}S_{n-2}}$ for ${\displaystyle n\geq 2}$

Generating function

The generating function ${\displaystyle G(x)}$ of the sequence ${\displaystyle (S_{n})_{n\geq 0}}$ is

${\displaystyle G(x)={\frac {1-x-{\sqrt {1-6x+x^{2}}}}{2x}}=\sum _{n=0}^{\infty }S_{n}x^{n}}$.^[7]

It can be expressed in terms of the generating function for Catalan numbers ${\displaystyle C(x)={\frac {1-{\sqrt {1-4x}}}{2x}}}$ as

${\displaystyle G(x)={\frac {1}{1-x}}C{\big (}{\frac {x}{(1-x)^{2}}}{\big )}.}$

Uses

One topic of combinatorics is tiling shapes, and one particular instance of this is domino tilings; the question in this instance is, "How many dominoes (that is, ${\displaystyle 1\times 2}$ or ${\displaystyle 2\times 1}$ rectangles) can we arrange on some shape such that none of the dominoes overlap, the entire shape is covered, and none of the dominoes stick out of the shape?" The shape that the Schröder numbers have a connection with is the Aztec diamond. Shown below for reference is an Aztec diamond of order 4 with a possible domino tiling.

It turns out that the determinant of the ${\displaystyle (2n-1)\times (2n-1)}$ Hankel matrix of the Schröder numbers, that is, the square matrix whose ${\displaystyle (i,j)}$th entry is ${\displaystyle S_{i+j-1},}$ is the number of domino tilings of the order ${\displaystyle n}$ Aztec diamond, which is ${\displaystyle 2^{n(n+1)/2}.}$^[9] That is,

${\displaystyle {\begin{vmatrix}S_{1}&S_{2}&\cdots &S_{n}\\S_{2}&S_{3}&\cdots &S_{n+1}\\\vdots &\vdots &\ddots &\vdots \\S_{n}&S_{n+1}&\cdots &S_{2n-1}\end{vmatrix}}=2^{n(n+1)/2}.}$

For example:

• ${\displaystyle {\begin{vmatrix}2\end{vmatrix}}=2=2^{1(2)/2}}$

• ${\displaystyle {\begin{vmatrix}2&6\\6&22\end{vmatrix}}=8=2^{2(3)/2}}$

• ${\displaystyle {\begin{vmatrix}2&6&22\\6&22&90\\22&90&394\end{vmatrix}}=64=2^{3(4)/2}}$

See also

• Delannoy number
• Motzkin number
• Narayana number
• Narayana polynomials
• Schröder–Hipparchus number
• Catalan number

References

1. Sloane, N. J. A. (ed.). "Sequence A006318 (Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 5 March 2018.
2. Ardila, Federico (2015). "Algebraic and geometric methods in enumerative combinatorics". Handbook of enumerative combinatorics. Boca Raton, FL: CRC Press. pp. 3–172.
3. Sloane, N. J. A. (ed.). "Sequence A001003 (Schroeder's second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 5 March 2018.
4. Drake, Dan (2010). "Bijections from weighted Dyck paths to Schröder paths". arXiv:1006.1959 [math.CO].
5. Deng, Eva Y. P.; Yan, Wei-Jun (2008). "Some identities on the Catalan, Motzkin, and Schröder numbers". Discrete Applied Mathematics. 156 (166–218X): 2781–2789. doi:10.1016/j.dam.2007.11.014.
6. Sloane, N. J. A. "Triangular array associated with Schroeder numbers". The On-Line Encyclopedia of Integer Sequences. Retrieved 5 March 2018.
7. Oi, Feng; Guo, Bai-Ni (2017). "Some explicit and recursive formulas of the large and little Schröder numbers". Arab Journal of Mathematical Sciences. 23 (1319–5166): 141–147. doi:10.1016/j.ajmsc.2016.06.002.
8. "Problem 4 (Solution)". IMC Problems 2019. IMC. Retrieved 2024-08-27.
9. Eu, Sen-Peng; Fu, Tung-Shan (2005). "A simple proof of the Aztec diamond theorem". Electronic Journal of Combinatorics. 12 (1077–8926): Research Paper 18, 8. doi:10.37236/1915. S2CID 5978643.

Further reading

• Weisstein, Eric W. "Schröder Number". MathWorld.
• Stanley, Richard P.: Catalan addendum to Enumerative Combinatorics, Volume 2