OFFSET
0,3
COMMENTS
As shown by Dulucq and Guilbert (see for example "Baxter Permutations", Discrete Math., 1998), a(n) is also the number of possible paths for three vicious walkers of length n-1 (a.k.a. "vicious 3-watermelons") [Essam & Guttmann (1995), Eq. (63)] [Jensen (2017), Eqs. (1), (2)]. The proof follows easily by comparing Ollerton's recurrence here and the recurrence in Eq. (60) of Essam & Guttmann. In fact, as discussed by Dulucq and Guilbert, this interpretation of the sequence has been known for a long time. - N. J. A. Sloane, Mar 19 2021; additional references provided by Olivier Gerard, Mar 22 2021.
From Roger Ford, Apr 11 2020: (Start)
a(n) is also the number of meanders with n top arches that as the number of arches is reduced by combining the first and last arches every even number of arches produces a meander.
Example: For n = 4, this meander has this trait.
/\ arches= 8
/\ / \
/ \ --> /\ //\ \
Starting meander: /\ /\ / /\ \ Split and /\/\//\\ ///\\/\\
\ \/ \ \/ / / rotate
\ \ / / bottom arches /\
\ \/ / / \ arches= 7
\ / / /\\ /\
\ / Combine start //\//\\\ //\\/\
\ / first arch with
meander: \/ end of last arch
/\ /\
/\ / \ Combine arches /\ //\\ arches= 6
\ \ / /\ \ again then /\ //\\ ///\\\
\ \ \/ / / rotate and join
\ \ / / <-- /\
\ \/ / //\\ /\ arches= 5
\ / Combine ///\\\ //\\
\/ /\
meander: / \
/ /\ \ Combine /\
\/ \/ <-- //\\ /\/\ arches= 4
/\
Combine /\ //\\ arches= 3
meander: /\ Combine
\/ <-- /\ /\ arches= 2
(End)
REFERENCES
Arthur T. Benjamin and Naiomi T. Cameron, Counting on determinants, American Mathematical Monthly, 112.6 (2005): 481-492.
W. M. Boyce, Generation of a class of permutations associated with commuting functions, Math. Algorithms, 2 (1967), 19-26.
William M. Boyce, Commuting functions with no common fixed point, Transactions of the American Mathematical Society 137 (1969): 77-92.
T. Y. Chow, Review of "Bonichon, Nicolas; Bousquet-Mélou, Mireille; Fusy, Éric; Baxter permutations and plane bipolar orientations. Sem. Lothar. Combin. 61A (2009/10), Art. B61Ah, 29 pp.", MathSciNet Review MR2734180 (2011m:05023).
S. Dulucq and O. Guibert, Baxter permutations, Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995).
J. W. Essam and A. J. Guttmann, Vicious walkers and directed polymer networks in general dimensions, Physical Review E, 52(6), 1995, pp. 5849ff. See (60) and (63).
Iwan Jensen, Three friendly walkers, Journal of Physics A: Mathematical and Theoretical, Volume 50:2 (2017), #24003, 14 pages; https://doi.org/10.1088/1751-8121/50/2/024003. See (1), (2).
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. See p. 399, Table A.7.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 Problem 6.55.
Doron Zeilberger, The method of creative telescoping, J. Symb. Comput. 11.3 (1991): 195-204. See Section 7.8.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1119 (first 101 terms from T. D. Noe)
Eyal Ackerman, Gill Barequet, and Ron Y. Pinter, On the number of rectangular partitions, SODA '04, 2004.
Andrei Asinowski, Gill Barequet, Mireille Bousquet-Mélou, Toufik Mansour, and Ron Y. Pinter, Orders induced by segments in floorplans and (2-14-3,3-41-2)-avoiding permutations, Elect. J. Combin. (2013) Vol. 20, No. 2, Art. P35. See also arXiv:1011.1889 [math.CO], 2010-2012.
Andrei Asinowski, Jean Cardinal, Stefan Felsner, and Éric Fusy, Combinatorics of rectangulations: Old and new bijections, arXiv:2402.01483 [math.CO], 2024. See p. 10.
Jean-Christophe Aval, Adrien Boussicault, Mathilde Bouvel, Olivier Guibert, and Matteo Silimbani, Baxter Tree-like Tableaux, arXiv:2108.06212 [math.CO], 2021.
Nicholas R. Beaton, Mathilde Bouvel, Veronica Guerrini, and Simone Rinaldi, Slicings of parallelogram polyominoes, or how Baxter and Schroeder can be reconciled, arXiv preprint arXiv:1511.04864 [math.CO], 2015.
Nicolas Bonichon, Mireille Bousquet-Mélou, and Éric Fusy, Baxter permutations and plane bipolar orientations, Sem. Lothar. Combin. 61A (2009/10), Art. B61Ah, 29 pp.
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
Alin Bostan, Jordan Tirrell, Bruce W. Westbury, and Yi Zhang, On sequences associated to the invariant theory of rank two simple Lie algebras, arXiv:1911.10288 [math.CO], 2019.
Alin Bostan, Jordan Tirrell, Bruce W. Westbury, and Yi Zhang, On some combinatorial sequences associated to invariant theory, arXiv:2110.13753 [math.CO], 2021.
Mireille Bousquet-Mélou, Four classes of pattern-avoiding permutations under one roof: generating trees with two labels, Electron. J. Combin. (2002/03), Vol. 9, No. 2, Art. R19, 31 pp.
Mathilde Bouvel, Veronica Guerrini, Andrew Rechnitzer, and Simone Rinaldi, Semi-Baxter and strong-Baxter: two relatives of the Baxter sequence, arXiv:1702.04529 [math.CO], 2017.
W. M. Boyce, Generation of a class of permutations associated with commuting functions, 1967; annotated and corrected scanned copy.
W. M. Boyce, Correction to page 25 (Part 1)
W. M. Boyce, Correction to Page 25 (Part 2)
W. M. Boyce, Baxter Permutations and Functional Composition, Houston J. Math. (1981) Vol. 7, No. 2.
Sophie Burrill, Stephen Melczer, and Marni Mishna, A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape, arXiv:1411.6606 [math.CO], 2014-2015.
Hal Canary, Aztec diamonds and Baxter permutations, arXiv:math/0309135 [math.CO], 2003-2004.
Jean Cardinal and Vincent Pilaud, Rectangulotopes, arXiv:2404.17349 [math.CO], 2024. See p. 14.
Grégory Chatel and Vincent Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015. and DOI.
Timothy Y. Chow, Henrik Eriksson, and C. Kenneth Fan, Chess tableaux, Elect. J. Combin., (2005) Vol. 11, No. 2, Art. A3.
Fan Rong K. Chung, Ronald L. Graham, Verner E. Hoggatt Jr., and M. Kleiman, The number of Baxter permutations J. Combin. Theory Ser. A (1978) Vol. 24, No. 3, 382-394.
James Cranch, Representing and Enumerating Two-Dimensional Pasting Diagrams, 2014.
CombOS - Combinatorial Object Server, Generate diagonal rectangulations
Colin Defant, Stack-Sorting Preimages of Permutation Classes, arXiv:1809.03123 [math.CO], 2018.
Anand Deopurkar, Eduard Duryev, and Anand Patel, Ramification divisors of general projections, arXiv:1901.01513 [math.AG], 2019.
Tomislav Došlic and Darko Veljan, Logarithmic behavior of some combinatorial sequences, Disc. Math. (2008) Vol. 308, No. 11, 2182-2212. MR2404544 (2009j:05019).
Serge Dulucq and Olivier Guibert, Permutations de Baxter, Séminaire Lotharingien de Combinatoire, B33c (1994), 9 pp.
Serge Dulucq and Olivier Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. (1996) Vol. 157, 91-106.
Serge Dulucq and Olivier Guibert, Baxter permutations, Disc. Math. (1998) Vol. 180, No. 1-3, 143-156. MR1603713 (99c:05004).
Hanqian Fang, Candice X. T. Zhang, and James J. Y. Zhao, Analytic properties arising from the Baxter numbers, arXiv:2505.05873 [math.CO], 2025. See pp. 2, 9.
Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A (2011) Vol. 118, No. 3, 993-1020.
Daniel C. Fielder and Cecil O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers (1990) Vol. 3, 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)
Daniel C. Fielder and Cecil O. Alford, On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles, Fib. Quart. (1989) Vol. 27, 160-168.
Paweł Gawrychowski and Patrick K. Nicholson, Optimal Encodings for Range Top-k, Selection, and Min-Max, arXiv:1411.6581 [cs.DS], 2014-2015.
Paweł Gawrychowski and Patrick K. Nicholson, Optimal Encodings for Range Top-k, Selection, and Min-Max, in Automata, Languages, and Programming, Lect. Notes Comp. Sci. (2015) Vol. 9134, 593-604.
Samuele Giraudo, Algebraic and combinatorial structures on Baxter permutations, DMTCS proc. AO, (FPSAC, Reykjavik, Iceland, 2011) 387-398.
Samuele Giraudo, Algebraic and combinatorial structures on pairs of twin binary trees, arXiv:1204.4776 [math.CO], 2012.
Samuele Giraudo, Algebraic and combinatorial structures on pairs of twin binary trees, J. Alg. (15 June 2012) Vol. 360, 115-157.
Olivier Guibert and Svante Linusson, Doubly alternating Baxter permutations are Catalan, Disc. Math. (2000) Vol 217, 157-166.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
Verner E. Hoggatt, Jr., Letter to N. J. A. Sloane, Apr 1977
Don Knuth, Twintrees, Baxter Permutations, and Floorplans, Christmas Lecture, Stanford 2022 (YouTube video).
Laszlo Kozma and T. Saranurak, Binary search trees and rectangulations, arXiv preprint arXiv:1603.08151 [cs.DS], 2016.
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 2.25].
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, arXiv:2103.09333 [math.CO], 2021.
Ahti Peder and Mati Tombak, Finding the description of structure by counting method: a case study, SOFSEM 2011, LNCS 6543 (2011) 455-466.
Vincent Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, arXiv:1505.07665 [math.CO], 2015.
Vic Reiner, Dennis Stanton, and Volkmar Welker, The Charney-Davis quantity for certain graded posets, Sem. Lothar. Combin. (2003/04) Vol. 50, Art. B50c, 13 pp.
Kaoru Sano, Rectangulations avoiding a pattern, arXiv:2511.22015 [math.CO], 2025.
Jannik Silvanus, Improved Cardinality Bounds for Rectangle Packing Representations, Doctoral Dissertation, Univ. Bonn (Rheinische Friedrich Wilhelms Universität, Germany 2019).
N. J. A. Sloane, Three vicious walkers of lengths 1, 2, and 3
Wikipedia, Baxter permutation
FORMULA
a(n) = Sum_{k=1..n} C(n+1,k-1) * C(n+1,k) * C(n+1,k+1) / (C(n+1,1) * C(n+1,2)).
(n + 1)*(n + 2)*(n + 3)*(3*n - 2)*a(n) = 2*(n + 1)*(9*n^3 + 3*n^2 - 4*n + 4)*a(n - 1) + (3*n - 1)*(n - 2)*(15*n^2 - 5*n - 14)*a(n - 2) + 8*(3*n + 1)*(n - 2)^2*(n - 3)*a(n - 3), if n>1. [Stanley, 1999] - Michael Somos, Jul 19 2002
From Richard L. Ollerton, Sep 13 2006: (Start)
D-finite with recurrence (n + 2)*(n + 3)*a(n) = (7*n^2 + 7*n - 2)*a(n-1) + 8*(n - 1)*(n - 2)*a(n-2), a(0) = a(1) = 1.
a(n) = hypergeom([-1 - n, -n, 1 - n], [2, 3], -1). (End)
[The coefficients of the polynomials p(n, x) = hypergeom([-1-n, -n, 1-n], [2, 3], -x) are given by the triangular form of A056939. - Peter Luschny, Dec 28 2022]
G.f.: -1 + (1/(3*x^2)) * (x-1 + (1-2*x)*hypergeom([-2/3, 2/3],[1],27*x^2/(1-2*x)^3) - (8*x^3-11*x^2-x)*hypergeom([1/3, 2/3],[2],27*x^2/(1-2*x)^3)/(1-2*x)^2 ). - Mark van Hoeij, Oct 23 2011
a(n) ~ 2^(3*n+5)/(Pi*sqrt(3)*n^4). - Vaclav Kotesovec, Oct 01 2012
0 = +a(n)*(+a(n+1)*(+512*a(n+2) + 2624*a(n+3) - 960*a(n+4)) +a(n+2)*(-1216*a(n+2) + 3368*a(n+3) - 1560*a(n+4)) + a(n+3)*(+600*a(n+3) + 120*a(n+4))) + a(n+1)*(+a(n+1)*(-64*a(n+2) - 1288*a(n+3) + 600*a(n+4)) +a(n+2)*(-136*a(n+2) + 295*a(n+3) - 421*a(n+4)) +a(n+3)*(+161*a(n+3) + 41*a(n+4))) + a(n+2)*(+a(n+2)*(+72*a(n+2) + 17*a(n+3) - 19*a(n+4)) + a(n+3)*(-a(n+3) - a(n+4))) if n>=0. - Michael Somos, Mar 09 2017
G.f.: ((x^3+3*x^2+3*x+1)/(1-8*x)^(3/4)*hypergeom([1/4, 5/4],[2],64*x*(1+x)^3/(8*x-1)^3) - 1 + x)/(3*x^2). - Mark van Hoeij, Nov 05 2023
EXAMPLE
G.f. = x + 2*x^2 + 6*x^3 + 22*x^4 + 92*x^5 + 422*x^6 + 2074*x^7 + ...
a(4) = 22 since all permutations of length 4 are Baxter except 2413 and 3142. - Michael Somos, Jul 19 2002
MAPLE
C := binomial; A001181 := proc(n) local k; add(C(n+1, k-1)*C(n+1, k)* C(n+1, k+1)/ (C(n+1, 1)*C(n+1, 2)), k = 1..n); end;
# Alternative:
a:= proc(n) option remember; `if`(n<2, 1,
((7*n^2+7*n-2)*a(n-1)+8*(n-1)*(n-2)*a(n-2))/((n+2)*(n+3)))
end:
seq(a(n), n=0..24); # Alois P. Heinz, Jul 29 2022
MATHEMATICA
A001181[n_] := HypergeometricPFQ[{-1-n, -n, 1-n}, {2, 3}, -1] (* Richard L. Ollerton, Sep 13 2006 *)
a[0]=1; a[1]=1; a[n_] := a[n] = ((7n^2+7n-2)*a[n-1] + 8(n-1)(n-2)*a[n-2]) / ((n+2)(n+3)); Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 28 2015, from 3rd formula *)
PROG
(PARI) {a(n) = if( n<0, 0, sum( k=1, n, binomial(n+1, k-1) * binomial(n+1, k) * binomial(n+1, k+1) / (binomial(n+1, 1) * binomial(n+1, 2))))}; /* Michael Somos, Jul 19 2002 */
(Haskell)
a001181 0 = 1
a001181 n =
(sum $ map (\k -> product $ map (a007318 (n+1)) [k-1..k+1]) [1..n])
`div` (a006002 n)
-- Reinhard Zumkeller, Oct 23 2011
(Python)
from sympy import binomial as C
def a(n): return sum([(C(n + 1, k - 1)*C(n + 1, k)*C(n + 1, k + 1))/(C(n + 1, 1) * C(n + 1, 2)) for k in range(1, n + 1)]) # Indranil Ghosh, Apr 25 2017
(Magma) [1] cat [2*(&+[Binomial(n+1, k-1)*Binomial(n+1, k)* Binomial(n+1, k+1): k in [1..n]])/(n*(n+1)^2): n in [1..30]]; // G. C. Greubel, Jul 24 2019
(SageMath) [1]+[2*sum(binomial(n+1, k-1)*binomial(n+1, k)*binomial(n+1, k+1) for k in (1..n))/(n*(n+1)^2) for n in (1..30)] # G. C. Greubel, Jul 24 2019
print([BaxterPermutations(n).cardinality() for n in range(25)])
# Peter Luschny, May 21 2024
(GAP) Concatenation([1], List([1..30], n-> 2*Sum([1..n], k-> Binomial(n+1, k-1)* Binomial(n+1, k)*Binomial(n+1, k+1) )/(n*(n+1)^2) )); # G. C. Greubel, Jul 24 2019
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Changed initial term to a(0) = 1 (it was a(0) = 0, but there were compelling reasons to change it). - N. J. A. Sloane, Sep 14 2021
STATUS
approved