The Schröder number is the number of lattice paths
in the Cartesian plane that start at (0, 0), end at , contain no points above the line , and are composed only of steps (0, 1), (1, 0), and (1,
1), i.e., ,
, and . The diagrams illustrating the paths generating , , and are illustrated above.
where ,
and the first few are 2, 6, 22, 90, ... (OEIS A006318).
The numbers of decimal digits in for , 1, ... are 1, 7, 74, 761, 7650, 76548, 765543, 7655504,
... (OEIS A114472), where the digits approach
those of
(OEIS A114491).
Bonin, J.; Shapiro, L.; and Simion, R. "Some -Analogs of the Schröder Numbers Arising from Combinatorial
Statistics on Lattice Paths." J. Stat. Planning Inference34,
35-55, 1993.Moser, L. and Zayachkowski, W. "Lattice Paths with
Diagonal Steps." Scripta Math.26, 223-229, 1963.Pergola,
E. and Sulanke, R. A. "Schröder Triangles, Paths, and Parallelogram
Polyominoes." J. Integer Sequences1, No. 98.1.7, 1998. http://www.math.uwaterloo.ca/JIS/VOL1/PergolaSulanke/.Rogers,
D. G. "A Schröder Triangle." Combinatorial Mathematics V:
Proceedings of the Fifth Australian Conference. New York: Springer-Verlag, pp. 175-196,
1977.Rogers, D. G. and Shapiro, L. "Some Correspondences involving
the Schröder Numbers." Combinatorial Mathematics: Proceedings of the
International Conference, Canberra, 1977. New York: Springer-Verlag, pp. 267-276,
1978.Schröder, E. "Vier kombinatorische Probleme." Z.
Math. Phys.15, 361-376, 1870.Sloane, N. J. A.
Sequences A006318/M1659, A114472,
and A114491 in "The On-Line Encyclopedia
of Integer Sequences."Stanley, R. P. "Hipparchus, Plutarch,
Schröder, Hough." Amer. Math. Monthly104, 344-350, 1997.Sulanke,
R. A. "Bijective Recurrences Concerning Schröder Paths." Electronic
J. Combinatorics5, No. 1, R47, 1-11, 1998. http://www.combinatorics.org/Volume_5/Abstracts/v5i1r47.html.