OFFSET
0,2
COMMENTS
For the little Schröder numbers (or little Schroeder numbers, or small Schroeder numbers) see A001003.
The number of perfect matchings in a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...). - Roberto E. Martinez II, Nov 05 2001
a(n) is the number of subdiagonal paths from (0, 0) to (n, n) consisting of steps East (1, 0), North (0, 1) and Northeast (1, 1) (sometimes called royal paths). - David Callan, Mar 14 2004
Twice A001003 (except for the first term).
a(n) is the number of dissections of a regular (n+4)-gon by diagonals that do not touch the base. (A diagonal is a straight line joining two nonconsecutive vertices and dissection means the diagonals are noncrossing though they may share an endpoint. One side of the (n+4)-gon is designated the base.) Example: a(1)=2 because a pentagon has only 2 such dissections: the empty one and the one with a diagonal parallel to the base. - David Callan, Aug 02 2004
a(n) is the number of separable permutations, i.e., permutations avoiding 2413 and 3142 (see Shapiro and Stephens). - Vincent Vatter, Aug 16 2006
Eric W. Weisstein comments that the Schröder numbers bear the same relationship to the Delannoy numbers (A001850) as the Catalan numbers (A000108) do to the binomial coefficients. - Jonathan Vos Post, Dec 23 2004
a(n) is the number of lattice paths from (0, 0) to (n+1, n+1) consisting of unit steps north N = (0, 1) and variable-length steps east E = (k, 0), with k a positive integer, that stay strictly below the line y = x except at the endpoints. For example, a(2) = 6 counts 111NNN, 21NNN, 3NNN, 12NNN, 11N1NN, 2N1NN (east steps indicated by their length). If the word "strictly" is replaced by "weakly", the counting sequence becomes the little Schröder numbers, A001003 (offset). - David Callan, Jun 07 2006
a(n) is the number of dissections of a regular (n+3)-gon with base AB that do not contain a triangle of the form ABP with BP a diagonal. Example: a(1) = 2 because the square D-C | | A-B has only 2 such dissections: the empty one and the one with the single diagonal AC (although this dissection contains the triangle ABC, BC is not a diagonal). - David Callan, Jul 14 2006
a(n) is the number of (colored) Motzkin n-paths with each upstep and each flatstep at ground level getting one of 2 colors and each flatstep not at ground level getting one of 3 colors. Example: With their colors immediately following upsteps/flatsteps, a(2) = 6 counts U1D, U2D, F1F1, F1F2, F2F1, F2F2. - David Callan, Aug 16 2006
The Hankel transform of this sequence is A006125(n+1) = [1, 2, 8, 64, 1024, 32768, ...]; example: Det([1, 2, 6, 22; 2, 6, 22, 90; 6, 22, 90, 394; 22, 90, 394, 1806]) = 64. - Philippe Deléham, Sep 03 2006
Triangle A144156 has row sums equal to A006318 with left border A001003. - Gary W. Adamson, Sep 12 2008
a(n) is also the number of order-preserving and order-decreasing partial transformations (of an n-chain). Equivalently, it is the order of the Schröder monoid, PC sub n. - Abdullahi Umar, Oct 02 2008
Sum_{n >= 0} a(n)/10^n - 1 = (9 - sqrt(41))/2. - Mark Dols, Jun 22 2010
1/sqrt(41) = Sum_{n >= 0} Delannoy number(n)/10^n. - Mark Dols, Jun 22 2010
a(n) is also the dimension of the space Hoch(n) related to Hochschild two-cocycles. - Ph. Leroux (ph_ler_math(AT)yahoo.com), Aug 24 2010
Let W = (w(n, k)) denote the augmentation triangle (as at A193091) of A154325; then w(n, n) = A006318(n). - Clark Kimberling, Jul 30 2011
Conjecture: For each n > 2, the polynomial sum_{k = 0}^n a(k)*x^{n-k} is irreducible modulo some prime p < n*(n+1). - Zhi-Wei Sun, Apr 07 2013
From Jon Perry, May 24 2013: (Start)
Consider a Pascal triangle variant where T(n, k) = T(n, k-1) + T(n-1, k-1) + T(n-1, k), i.e., the order of performing the calculation must go from left to right (A033877). This sequence is the rightmost diagonal.
Triangle begins:
1;
1, 2;
1, 4, 6;
1, 6, 16, 22;
1, 8, 30, 68, 90;
... (End)
a(n) is the number of permutations avoiding 2143, 3142 and one of the patterns among 246135, 254613, 263514, 524361, 546132. - Alexander Burstein, Oct 05 2014
a(n) is the number of semi-standard Young tableaux of shape n x 2 with consecutive entries. That is, j in P and 1 <= i<= j imply i in P. - Graham H. Hawkes, Feb 15 2015
a(n) is the number of unary-rooted size n unary-binary trees (each node has either 1 or 2 degree out). - John Bodeen, May 29 2017
Conjecturally, a(n) is the number of permutations pi of length n such that s(pi) avoids the patterns 231 and 321, where s denotes West's stack-sorting map. - Colin Defant, Sep 17 2018
a(n) is the number of n X n permutation matrices which percolate under the 2-neighbor bootstrap percolation rule (see Shapiro and Stephens). The number of general n X n matrices of weight n which percolate is given in A146971. - Jonathan Noel, Oct 05 2018
a(n) is the number of permutations of length n+1 which avoid 3142 and 3241. The permutations are precisely the permutations that are sortable by a decreasing stack followed by an increasing stack in series. - Rebecca Smith, Jun 06 2019
a(n) is the number of permutations of length n+1 avoiding the partially ordered pattern (POP) {3>1, 4>1, 1>2} of length 4. That is, the number of length n+1 permutations having no subsequences of length 4 in which the second element is the smallest, and the first element is smaller than the third and fourth elements. - Sergey Kitaev, Dec 10 2020
Named after the German mathematician Ernst Schröder (1841-1902). - Amiram Eldar, Apr 15 2021
a(n) is the number of sequences of nonnegative integers (u_1, u_2, ..., u_n) such that (i) u_i <= i for all i, and (ii) the nonzero u_i are weakly increasing. For example, a(2) = 6 counts 00, 01, 02, 10, 11, 12. See link "Some bijections for lattice paths" at A001003. - David Callan, Dec 18 2021
a(n) is the number of separable elements of the Weyl group of type B_n/C_n (see Gaetz and Gao). - Fern Gossow, Jul 31 2023
The number of domino tilings of an Aztec triangle of order n. Dually, the number perfect matchings of the edges in the cellular graph formed by a triangular grid of n squares (n = 1, 4, 9, 16, 25, ...) as in Ciucu (1996). - Michael Somos, Sep 16 2024
REFERENCES
D. Andrica and E. J. Ionascu, On the number of polynomials with coefficients in [n], An. St. Univ. Ovidius Constanta, 2013, to appear.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Riordan-Bernstein Polynomials, Hankel Transforms and Somos Sequences, Journal of Integer Sequences, Vol. 15 2012, #12.8.2.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
O. Bodini, A. Genitrini, F. Peschanski, and N.Rolin, Associativity for binary parallel processes, CALDAM 2015.
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 24, 618.
S. Brlek, E. Duchi, E. Pergola, and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
Xiang-Ke Chang, XB Hu, H Lei, and YN Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
William Y. C. Chen and Carol J. Wang, Noncrossing Linked Partitions and Large (3, 2)-Motzkin Paths, Discrete Math., 312 (2012), 1918-1922.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 81, #21, (4), q_n.
D. E. Davenport, L. W. Shapiro, and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012), #P33.
Deng, Eva Y. P.; Dukes, Mark; Mansour, Toufik; and Wu, Susan Y. J.; Symmetric Schröder paths and restricted involutions. Discrete Math. 309 (2009), no. 12, 4108-4115. See p. 4109.
E. Deutsch, A bijective proof of an equation linking the Schroeder numbers, large and small, Discrete Math., 241 (2001), 235-240.
C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
Doslic, Tomislav and Veljan, Darko. Logarithmic behavior of some combinatorial sequences. Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019) - From N. J. A. Sloane, May 01 2012
M. Dziemianczuk, Generalizing Delannoy numbers via counting weighted lattice paths, INTEGERS, 13 (2013), #A54.
Egge, Eric S., Restricted signed permutations counted by the Schröder numbers. Discrete Math. 306 (2006), 552-563. [Many applications of these numbers.]
S. Getu et al., How to guess a generating function, SIAM J. Discrete Math., 5 (1992), 497-499.
S. Gire, Arbres, permutations a motifs exclus et cartes planaire: quelques problemes algorithmiques et combinatoires, Ph.D. Thesis, Universite Bordeaux I, 1993.
N. S. S. Gu, N. Y. Li, and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
Guruswami, Venkatesan, Enumerative aspects of certain subclasses of perfect graphs. Discrete Math. 205 (1999), 97-117.
Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
D. E. Knuth, The Art of Computer Programming, Vol. 1, Section 2.2.1, Problem 11.
D. Kremer, Permutations with forbidden subsequences and a generalized Schröder number, Discrete Math. 218 (2000) 121-130.
Kremer, Darla and Shiu, Wai Chee; Finite transition matrices for permutations avoiding pairs of length four patterns. Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
Laradji, A. and Umar, A. Asymptotic results for semigroups of order-preserving partial transformations. Comm. Algebra 34 (2006), 1071-1075. - Abdullahi Umar, Oct 11 2008
L. Moser and W. Zayachkowski, Lattice paths with diagonal steps, Scripta Math., 26 (1961), 223-229.
L. Shapiro and A. B. Stephens, Bootstrap percolation, the Schröder numbers and the N-kings problem, SIAM J. Discrete Math., Vol. 4 (1991), pp. 275-280.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 178 and also Problems 6.39 and 6.40.
Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
Sheng-Liang Yang and Mei-yang Jiang, The m-Schröder paths and m-Schröder numbers, Disc. Math. (2021) Vol. 344, Issue 2, 112209. doi:10.1016/j.disc.2020.112209. See Table 1.
LINKS
Fung Lam, Table of n, a(n) for n = 0..2000 (terms 0..100 by T. D. Noe)
J. Abate and W. Whitt, Integer Sequences from Queueing Theory, J. Int. Seq. 13 (2010), 10.5.5, Corollary 8.
M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008), 2544-2563.
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Andrei Asinowski, G. Barequet, M. Bousquet-Mélou, T. Mansour, and R. Pinter, Orders induced by segments in floorplans and (2-14-3,3-41-2)-avoiding permutations, arXiv:1011.1889 [math.CO], 2010-2012.
M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Preprint, 2002.
M. D. Atkinson and T. Stitt, Restricted permutations and the wreath product, Discrete Math., 259 (2002), 19-36.
Yu Hin Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Jean-Christophe Aval and F. Bergeron, Rectangular Schroder Parking Functions Combinatorics, arXiv:1603.09487 [math.CO], 2016.
Axel Bacher, Improving the Florentine algorithms: recovering algorithms for Motzkin and Schröder paths, arXiv:1802.06030 [cs.DS], 2018.
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
Kayleigh Bangs, Skye Binegar, Young Kim, Kyle Ormsby, Angélica M. Osorno, David Tamas-Parris, and Livia Xu, Biased permutative equivariant categories, arXiv:1907.00933 [math.AT], 2019.
E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, Permutations avoiding an increasing number of length-increasing forbidden subsequences, Discrete Mathematics and Theoretical Computer Science, 4 (2000), 31-44.
E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, Some permutations with forbidden subsequences and their inversion number, Discrete Math. 234(1-3) (2001), 1-15.
E. Barcucci, E. Pergola, R. Pinzani, and S. Rinaldi, ECO method and hill-free generalized Motzkin paths, Séminaire Lotharingien de Combinatoire, B46b (2001), 14 pp.
J.-L. Baril, C. Khalil, and V. Vajnovszki, Catalan and Schröder permutations sortable by two restricted stacks, arXiv:2004.01812 [cs.DM], 2020.
Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, Motzkin and Catalan Tunnel Polynomials, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6
Paul Barry, Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences , J. Int. Seq. 13 (2010) #10.7.2.
Paul Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011), Article 11.4.5.
Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., 22 (2019), Article 19.5.8.
Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.
P. Barry and A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 11.3.8.
Paul Barry and Nikolaos Pantelidis,On pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays, J Algebr Comb 54, 399-423 (2021). (appeared in its aerated form,i.e. 1,0,2,0,6,0,...)
Christian Bean, Émile Nadeau, and Henning Ulfarsson, Enumeration of Permutation Classes and Weighted Labelled Independent Sets, arXiv:1912.07503 [math.CO], 2019.
Arkady Berenstein, Vladimir Retakh, Christophe Reutenauer, and Doron Zeilberger, The Reciprocal of Sum_{n >= 0} a^n b^n for non-commuting a and b, Catalan numbers and non-commutative quadratic equations, arXiv preprint arXiv:1206.4225 [math.CO], 2012. - From N. J. A. Sloane, Nov 28 2012
F. R. Bernhart and N. J. A. Sloane, Emails, April-May 1994.
J. Bloom and A. Burstein, Egge triples and unbalanced Wilf-equivalence, arXiv:1410.0230 [math.CO], 2014.
O. Bodini, A. Genitrini, and F. Peschanski, The Combinatorics of Non-determinism, In proc. IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS'13), Leibniz International Proceedings in Informatics, pp 425-436, 2013.
Miklós Bóna, Cheyne Homberger, Jay Pantone, and Vince Vatter, Pattern-avoiding involutions: exact and asymptotic enumeration, arxiv:1310.7003 [math.CO], 2013-2014.
M. Bremner and S. Madariaga, Lie and Jordan products in interchange algebras, arXiv:1408.3069 [math.RA], 2014-2015.
M. Bremner and S. Madariaga, Permutation of elements in double semigroups, arXiv:1405.2889 [math.RA], 2014-2015.
R. Brignall, S. Huczynska, and V. Vatter, Simple permutations and algebraic generating functions, arXiv:math/0608391 [math.CO], 2006.
Marie-Louise Bruner and Martin Lackner, On the Likelihood of Single-Peaked Preferences, arXiv:1505.05852 [cs.GT], 2015.
Alexander Burstein, Sergi Elizalde, and Toufik Mansour, Restricted Dumont permutations, Dyck paths and noncrossing partitions, arXiv:math/0610234 [math.CO], 2006. See Theorem 3.5.
Alexander Burstein and J. Pantone, Two examples of unbalanced Wilf-equivalence, arXiv:1402.3842 [math.CO], 2014.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
David Callan, An application of a bijection of Mansour, Deng, and Du, arXiv:1210.6455 [math.CO], 2012.
David Callan, A note on a bijection for Schröder permutations, arXiv:1602.05571 [math.CO], 2016.
David Callan and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.
Hui-Qin Cao and Hao Pan, A Stern-type congruence for the Schröder numbers, arXiv:1512.06310 [math.NT], 2015.
Jean Cardinal, Vera Sacristán, and Rodrigo I. Silveira, A Note on Flips in Diagonal Rectangulations, arXiv:1712.07919 [math.CO], 2017.
F. Chapoton, F. Hivert, and J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv:1307.0092 [math.CO], 2013.
F. Chapoton and S. Giraudo, Enveloping operads and bicoloured noncrossing configurations, arXiv:1310.4521 [math.CO], 2013.
W. Y. C. Chen, L. H. Liu, and C. J. Wang, Linked Partitions and Permutation Tableaux, arXiv:1305.5357 [math.CO], 2013.
Z. Chen and H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 (2016), eq. (1.13), a=2, b=1.
Shane Chern, On 0012-avoiding inversion sequences and a Conjecture of Lin and Ma, arXiv:2006.04318 [math.CO], 2020.
J. Cigler, Hankel determinants of some polynomial sequences, 2012.
Johann Cigler, Christian Krattenthaler, Hankel determinants of linear combinations of moments of orthogonal polynomials, arXiv:2003.01676 [math.CO], 2020.
M. Ciucu, Perfect matchings of cellular graphs, J. Algebraic Combin., 5 (1996) 87-103.
CombOS - Combinatorial Object Server, Generate slicing floorplans
Sylvie Corteel, Megan A. Martinez, Carla D. Savage, and Michael Weselcouch, Patterns in Inversion Sequences I, arXiv:1510.05434 [math.CO], 2015
S. Crowley, Integral Transforms of the Harmonic Sawtooth Map, The Riemann Zeta Function, Fractal Strings, and a Finite Reflection Formula, arXiv:1210.5652 [math.NT], 2012.
S. Crowley, Mellin and Laplace Integral Transforms Related to the Harmonic Sawtooth Map and a Diversion Into The Theory Of Fractal Strings, Number Theory, viXra:1202.0079, 2012.
R. De Castro, A. L. Ramírez, and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, arXiv:1310.2449 [math.PR], 2013.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Colin Defant, Troupes, Cumulants, and Stack-Sorting, arXiv:2004.11367 [math.CO], 2020.
Phan Thuan Do, Thi Thu Huong Tran, and Vincent Vajnovszki, Exhaustive generation for permutations avoiding a (colored) regular sets of patterns, arXiv:1809.00742 [cs.DM], 2018.
B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths (Example 1.6.7), A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
D. Drake, Bijections from Weighted Dyck Paths to Schröder Pathss</a, J. Int. Seq. 13 (2010), #10.9.2.
Rosena R. X. Du, Xiaojie Fan, and Yue Zhao, Enumeration on row-increasing tableaux of shape 2 X n, arXiv:1803.01590 [math.CO], 2018.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
Ömer Eğecioğlu, Collier Gaiser, and Mei Yin, Enumerating pattern-avoiding permutations by leading terms, arXiv:2309.15964 [math.CO], 2023.
S.-P. Eu and T.-S. Fu, A simple proof of the Aztec diamond problem, arXiv:math/0412041 [math.CO], 2004.
Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, arXiv:1203.6792 [math.CO], 2012 and J. Int. Seq. 17 (2014), #14.1.5
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see p. 474.
Shishuo Fu, Z. Lin, and J. Zeng, Two new unimodal descent polynomials, arXiv:1507.05184 [math.CO], 2015-2019.
Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
Christian Gaetz and Yibo Gao, Separable elements in Weyl groups, arXiv:1905.09331 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Olivier Gérard, Illustration of initial terms
Étienne Ghys, Quand beaucoup de courbes se rencontrent — Images des Mathématiques, CNRS, 2009.
Étienne Ghys, Intersecting curves, Amer. Math. Monthly, 120 (2013), 232-242.
Étienne Ghys, A Singular Mathematical Promenade, arXiv:1612.06373, 2016.
Samuele Giraudo, Operads from posets and Koszul duality, arXiv:1504.04529 [math.CO], 2015.
Samuele Giraudo, Pluriassociative algebras II: The polydendriform operad and related operads, arXiv:1603.01394 [math.CO], 2016.
Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019.
D. Gouyou-Beauchamps and B. Vauquelin, Deux propriétés combinatoires des nombres de Schröder, Theor. Inform. Appl., 22 (1988), 361-388.
Li Guo and Jun Pei, Averaging algebras, Schröder numbers and rooted trees, arXiv:1401.7386 [math.RA], 2014.
Nils Haug, T. Prellberg, and G. Siudem, Scaling in area-weighted generalized Motzkin paths, arXiv:1605.09643 [cond-mat.stat-mech], 2016.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, 1410.2657 [math.CO], 2014.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 159.
S. Kamioka, Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds, arXiv:1309.0268 [math.CO], 2013.
Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv:1201.1323 [math.CO], 2012.
S. Kitaev and J. Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012), #12.4.7
Laszlo Kozma and T. Saranurak, Binary search trees and rectangulations, arXiv:1603.08151 [cs.DS], 2016.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
G. Kreweras, Aires des chemins surdiagonaux et application à un problème économique, Cahiers du Bureau universitaire de recherche opérationnelle Série Recherche 24 (1976): 1-8. [Annotated scanned copy]
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, 8 (2005), Article 05.5.5.
Guillaume Lample and François Charton, Deep Learning for Symbolic Mathematics, arXiv:1912.01412 [cs.SC], 2019.
A. Laradji and A. Umar Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra, 278, (2004), 342-359.
A. Laradji and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), #04.3.8.
Tamás Lengyel, On some p-adic properties and supercongruences of Delannoy and Schröder Numbers, Integers (2021) Vol. 21, #A86.
Philippe Leroux, Hochschild two-cocycles and the good triple (As,Hoch,Mag infinity), arXiv:0806.4093 [math.RA], 2008.
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 5.
Huyile Liang, Jeffrey Remmel, and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see pp. 18-19.
Peter Luschny, The Lost Catalan Numbers And The Schröder Tableaux.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016 [Section 2.24].
Peter McCalla and Asamoah Nkwanta, Catalan and Motzkin Integral Representations, arXiv:1901.07092 [math.NT], 2019.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
J.-C. Novelli and J.-Y. Thibon, Hopf algebras and dendriform structures arising from parking functions, Fundamenta Mathematicae 193 (2007), no. 3, 189-241 (arXiv:math/0511200 [math.CO]).
Igor Pak, Complexity problems in enumerative combinatorics, arXiv:1803.06636 [math.CO], 2018.
P. Peart and W.-J. Woan, Generating Functions via Hankel and Stieltjes Matrices, J. Integer Seqs., Vol. 3 (2000), #00.2.1.
Jun Pei and Li Guo, Averaging algebras, Schröder numbers, rooted trees and operads, Journal of Algebraic Combinatorics, 42(1) (2015), 73-109; arXiv:1401.7386 [math.RA], 2014.
E. Pergola and R. A. Sulanke, Schröder Triangles, Paths and Parallelogram Polyominoes, J. Integer Sequences, 1 (1998), #98.1.7.
Feng Qi and B.-N. Guo, Some explicit and recursive formulas of the large and little Schröder numbers, Arab Journal of Mathematical Sciences, June 2016.
Feng Qi, Xiao-ting Shi, and Bai-Ni Guo, Integral representations of the large and little Schroder numbers, preprint, 2016.
Markus Saers, Dekai Wu, and Chris Quirk, On the Expressivity of Linear Transductions, The 13th Machine Translation Summit.
Seunghyun Seo, The Catalan Threshold Arrangement, Journal of Integer Sequences, 20 (2017), #17.1.1.
L. W. Shapiro and N. J. A. Sloane, Correspondence, 1976.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1978), 261-272.
P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers [Corrected annotated scanned copy]
R. A. Sulanke, Moments of generalized Motzkin paths, J. Integer Sequences, Vol. 3 (2000), #00.1.
R. A. Sulanke, Bijective recurrences concerning Schröder paths, Electron. J. Combin. 5 (1998), Research Paper 47, 11 pp.
Hua Sun and Yi Wang, A Combinatorial Proof of the Log-Convexity of Catalan-Like Numbers, J. Int. Seq. 17 (2014), #14.5.2
Zhi-Wei Sun, On Delannoy numbers and Schröder numbers, Journal of Number Theory, 131(12), (2011), 2387-2397; doi:10.1016/j.jnt.2011.06.005; arXiv 1009.2486 [math.NT].
Zhi-Wei Sun, Conjectures involving combinatorial sequences, arXiv:1208.2683 [math.CO], 2012. - N. J. A. Sloane, Dec 25 2012
Z.-W. Sun, Conjectures involving arithmetical sequences, Number Theory: Arithmetic in Shangri-La (eds., S. Kanemitsu, H.-Z. Li and J.-Y. Liu), Proc. the 6th China-Japan Sem. Number Theory (Shanghai, August 15-17, 2011), World Sci., Singapore, 2013, pp. 244-258. - N. J. A. Sloane, Dec 28 2012
Paul Tarau, On Type-directed Generation of Lambda Terms, preprint, 2015.
Paul Tarau, On logic programming representations of lambda terms: de Bruijn indices, compression, type inference, combinatorial generation, normalization, 2015.
Paul Tarau, A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations, arXiv:1507.06944 [cs.LO], 2015.
Paul Tarau, A Hiking Trip Through the Orders of Magnitude: Deriving Efficient Generators for Closed Simply-Typed Lambda Terms and Normal Forms, arXiv preprint arXiv:1608.03912 [cs.PL], 2016.
V. K. Varma and H. Monien, Renormalization of two-body interactions due to higher-body interactions of lattice bosons, arXiv:1211.5664 [cond-mat.quant-gas], 2012. - N. J. A. Sloane, Jan 03 2013
Vincent Vatter, Permutation classes, arXiv:1409.5159 [math.CO], 2014.
Yi Wang and Bao-Xuan Zhu, Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences, arXiv:1303.5595 [math.CO], 2013.
M. S. Waterman, Home Page (contains copies of his papers)
Eric Weisstein's World of Mathematics, Schröder Number.
J. West, Generating trees and the Catalan and Schröder numbers, Discrete Math. 146 (1995), 247-262.
J. Winter, M. M. Bonsangue, and J. J. M. M. Rutten, Context-free coalgebras, 2013.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.
S.-n. Zheng and S.-l. Yang, On the-Shifted Central Coefficients of Riordan Matrices, Journal of Applied Mathematics 2014, Article ID 848374.
FORMULA
G.f.: (1 - x - (1 - 6*x + x^2)^(1/2))/(2*x).
a(n) = 2*hypergeom([-n+1, n+2], [2], -1). - Vladeta Jovovic, Apr 24 2003
For n > 0, a(n) = (1/n)*Sum_{k = 0..n} 2^k*C(n, k)*C(n, k-1). - Benoit Cloitre, May 10 2003
The g.f. satisfies (1 - x)*A(x) - x*A(x)^2 = 1. - Ralf Stephan, Jun 30 2003
For the asymptotic behavior, see A001003 (remembering that A006318 = 2*A001003). - N. J. A. Sloane, Apr 10 2011
From Philippe Deléham, Nov 28 2003: (Start)
a(n) = Sum_{k = 0..n} C(n+k, n)*C(n, k)/(k+1). (End)
With offset 1: a(1) = 1, a(n) = a(n-1) + Sum_{i = 1..n-1} a(i)*a(n-i). - Benoit Cloitre, Mar 16 2004
a(n) = Sum_{k = 0..n} A000108(k)*binomial(n+k, n-k). - Benoit Cloitre, May 09 2004
a(n) = Sum_{k = 0..n} A011117(n, k). - Philippe Deléham, Jul 10 2004
a(n) = (CentralDelannoy(n+1) - 3 * CentralDelannoy(n))/(2*n) = (-CentralDelannoy(n+1) + 6 * CentralDelannoy(n) - CentralDelannoy(n-1))/2 for n >= 1, where CentralDelannoy is A001850. - David Callan, Aug 16 2006
From Abdullahi Umar, Oct 11 2008: (Start)
and 2*A123164(n) = (n+1)*a(n) - (n-1)*a(n-1) (n > 0). (End)
Define the general Delannoy numbers d(i, j) as in A001850. Then a(k) = d(2*k, k) - d(2*k, k-1) and a(0) = 1, Sum_{j=0..n} ((-1)^j * (d(n, j) + d(n-1, j-1)) * a(n-j)) = 0. - Peter E John, Oct 19 2006
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example, Phi([1]) is the Catalan numbers A000108. The present sequence is (essentially) Phi([2]). - Gary W. Adamson, Oct 27 2008
G.f.: 1/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x/(1-2x/(1-x.... (continued fraction). - Paul Barry, Dec 08 2008
G.f.: 1/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - x - x/(1 - ... (continued fraction). - Paul Barry, Jan 29 2009
a(n) ~ ((3 + 2*sqrt(2))^n)/(n*sqrt(2*Pi*n)*sqrt(3*sqrt(2) - 4))*(1-(9*sqrt(2) + 24)/(32*n) + ...). - G. Nemes (nemesgery(AT)gmail.com), Jan 25 2009
Logarithmic derivative yields A002003. - Paul D. Hanna, Oct 25 2010
a(n) = the upper left term in M^(n+1), M = the production matrix:
1, 1, 0, 0, 0, 0, ...
1, 1, 1, 0, 0, 0, ...
2, 2, 1, 1, 0, 0, ...
4, 4, 2, 1, 1, 0, ...
8, 8, 8, 2, 1, 1, ...
... - Gary W. Adamson, Jul 08 2011
a(n) is the sum of top row terms in Q^n, Q = an infinite square production matrix as follows:
1, 1, 0, 0, 0, 0, ...
1, 1, 2, 0, 0, 0, ...
1, 1, 1, 2, 0, 0, ...
1, 1, 1, 1, 2, 0, ...
1, 1, 1, 1, 1, 2, ...
... - Gary W. Adamson, Aug 23 2011
From Tom Copeland, Sep 21 2011: (Start)
With F(x) = (1 - 3*x - sqrt(1 - 6*x + x^2))/(2*x) an o.g.f. (nulling the n = 0 term) for A006318, G(x) = x/(2 + 3*x + x^2) is the compositional inverse.
Consequently, with H(x) = 1/ (dG(x)/dx) = (2 + 3*x + x^2)^2 / (2 - x^2),
a(n) = (1/n!)*[(H(x)*d/dx)^n] x evaluated at x = 0, i.e.,
F(x) = exp[x*H(u)*d/du] u, evaluated at u = 0. Also, dF(x)/dx = H(F(x)). (End)
a(n-1) = number of ordered complete binary trees with n leaves having k internal vertices colored black, the remaining n - 1 - k internal vertices colored white, and such that each vertex and its rightmost child have different colors ([Drake, Example 1.6.7]). For a refinement of this sequence see A175124. - Peter Bala, Sep 29 2011
D-finite with recurrence: (n-2)*a(n-2) - 3*(2*n-1)*a(n-1) + (n+1)*a(n) = 0. - Vaclav Kotesovec, Oct 05 2012
G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (1 - G(0))/x; G(k) = 1 + x - 2*x/G(k+1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 04 2012
G.f.: A(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) = (G(0) - 1)/x; G(k) = 1 - x/(1 - 2/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Jan 04 2012
a(n+1) = a(n) + Sum_{k=0..n} a(k)*(n-k). - Reinhard Zumkeller, Nov 13 2012
G.f.: 1/Q(0) where Q(k) = 1 + k*(1 - x) - x - x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013
a(-1-n) = a(n). - Michael Somos, Apr 03 2013
G.f.: 1/x - 1 - U(0)/x, where U(k) = 1 - x - x/U(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
G.f.: (2 - 2*x - G(0))/(4*x), where G(k) = 1 + 1/( 1 - x*(6 - x)*(2*k - 1)/(x*(6 - x)*(2*k - 1) + 2*(k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013
a(n) = 1/(n + 1) * (Sum_{j=0..n} C(n+j, j)*C(n+j+1, j+1)*(Sum_{k=0..n-j} (-1)^k*C(n+j+k, k))). - Graham H. Hawkes, Feb 15 2015
a(n) = hypergeom([-n, n+1], [2], -1). - Peter Luschny, Mar 23 2015
a(n) = sqrt(2) * LegendreP(n, -1, 3) where LegendreP is the associated Legendre function of the first kind (in Maple's notation). - Robert Israel, Mar 23 2015
G.f. A(x) satisfies: A(x) = Sum_{j>=0} x^j * Sum_{k=0..j} binomial(j,k)*A(x)^k. - Ilya Gutkovskiy, Apr 11 2019
From Peter Bala, May 13 2024: (Start)
a(n) = 2 * Sum_{k = 0..floor(n/2)} binomial(n, 2*k)*binomial(2*n-2*k, n)/(n-2*k+1) for n >= 1.
a(n) = Integral_{x = 0..1} Legendre_P(n, 2*x+1) dx. (End)
G.f. A(x) = 1/(1 - x) * c(x/(1-x)^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. - Peter Bala, Aug 29 2024
EXAMPLE
a(3) = 22 since the top row of Q^n = (6, 6, 6, 4, 0, 0, 0, ...); where 22 = (6 + 6 + 6 + 4).
G.f. = 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + 1806*x^6 + 8858*x^7 + 41586*x^8 + ...
MAPLE
Order := 24: solve(series((y-y^2)/(1+y), y)=x, y); # then A(x)=y(x)/x
BB:=(-1-z-sqrt(1-6*z+z^2))/2: BBser:=series(BB, z=0, 24): seq(coeff(BBser, z, n), n=1..23); # Zerinvary Lajos, Apr 10 2007
A006318_list := proc(n) local j, a, w; a := array(0..n); a[0] := 1;
for w from 1 to n do a[w] := 2*a[w-1]+add(a[j]*a[w-j-1], j=1..w-1) od; convert(a, list)end: A006318_list(22); # Peter Luschny, May 19 2011
A006318 := n-> add(binomial(n+k, n-k) * binomial(2*k, k)/(k+1), k=0..n): seq(A006318(n), n=0..22); # Johannes W. Meijer, Jul 14 2013
seq(simplify(hypergeom([-n, n+1], [2], -1)), n=0..100); # Robert Israel, Mar 23 2015
MATHEMATICA
a[0] = 1; a[n_Integer] := a[n] = a[n - 1] + Sum[a[k]*a[n - 1 - k], {k, 0, n - 1}]; Array[a[#] &, 30]
InverseSeries[Series[(y - y^2)/(1 + y), {y, 0, 24}], x] (* then A(x) = y(x)/x *) (* Len Smiley, Apr 11 2000 *)
CoefficientList[Series[(1 - x - (1 - 6x + x^2)^(1/2))/(2x), {x, 0, 30}], x] (* Harvey P. Dale, May 01 2011 *)
a[ n_] := 2 Hypergeometric2F1[ -n + 1, n + 2, 2, -1]; (* Michael Somos, Apr 03 2013 *)
a[ n_] := With[{m = If[ n < 0, -1 - n, n]}, SeriesCoefficient[(1 - x - Sqrt[ 1 - 6 x + x^2])/(2 x), {x, 0, m}]]; (* Michael Somos, Jun 10 2015 *)
Table[-(GegenbauerC[n+1, -1/2, 3] + KroneckerDelta[n])/2, {n, 0, 30}] (* Vladimir Reshetnikov, Nov 12 2016 *)
PROG
(PARI) {a(n) = if( n<0, n = -1-n); polcoeff( (1 - x - sqrt( 1 - 6*x + x^2 + x^2 * O(x^n))) / 2, n+1)}; /* Michael Somos, Apr 03 2013 */
(PARI) {a(n) = if( n<1, 1, sum( k=0, n, 2^k * binomial( n, k) * binomial( n, k-1)) / n)};
(Sage) # Generalized algorithm of L. Seidel
def A006318_list(n) :
D = [0]*(n+1); D[1] = 1
b = True; h = 1; R = []
for i in range(2*n) :
if b :
for k in range(h, 0, -1) : D[k] += D[k-1]
h += 1;
else :
for k in range(1, h, 1) : D[k] += D[k-1]
R.append(D[h-1]);
b = not b
return R
A006318_list(23) # Peter Luschny, Jun 02 2012
(Haskell)
a006318 n = a004148_list !! n
a006318_list = 1 : f [1] where
f xs = y : f (y : xs) where
y = head xs + sum (zipWith (*) xs $ reverse xs)
-- Reinhard Zumkeller, Nov 13 2012
(Python)
from gmpy2 import divexact
A006318 = [1, 2]
for n in range(3, 10**3):
# Chai Wah Wu, Sep 01 2014
(GAP) Concatenation([1], List([1..25], n->(1/n)*Sum([0..n], k->2^k*Binomial(n, k)*Binomial(n, k-1)))); # Muniru A Asiru, Nov 29 2018
KEYWORD
nonn,easy,core,nice
AUTHOR
EXTENSIONS
Edited by Charles R Greathouse IV, Apr 20 2010
STATUS
approved