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A001181 Number of Baxter permutations of length n (also called Baxter numbers).
(Formerly M1661 N0652)
33
1, 1, 2, 6, 22, 92, 422, 2074, 10754, 58202, 326240, 1882960, 11140560, 67329992, 414499438, 2593341586, 16458756586, 105791986682, 687782586844, 4517543071924, 29949238543316, 200234184620736, 1349097425104912, 9154276618636016, 62522506583844272 (list; graph; refs; listen; history; text; internal format)
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

Benjamin, Arthur T., and Naiomi T. Cameron. "Counting on determinants." The 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.

Boyce, William M. "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).

Dulucq, S.; Guibert, O. Baxter permutations. Proceedings of the 7th Conference on Formal Power Series and Algebraic Combinatorics (Noisy-le-Grand, 1995).

Essam, J. W. and Guttmann, A. J., 1995. Vicious walkers and directed polymer networks in general dimensions. Physical Review E, 52(6), 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.

Zeilberger, Doron. "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)

E. Ackerman et al., On the number of rectangular partitions, SODA '04, 2004.

A. 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, Electronic Journal of Combinatorics, 20:2 (2013), Paper P35; also arXiv preprint arXiv:1011.1889 [math.CO], 2010-2012.

Jean-Christophe Aval, Adrien Boussicault, Mathilde Bouvel, Olivier Guibert and Matteo Silimbani, Baxter Tree-like Tableaux, arXiv:2108.06212 [math.CO], 2021.

N. R. Beaton, M. Bouvel, V. Guerrini and S. 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.

F. Bousquet-Mélou, Four classes of pattern-avoiding permutations under one roof: generating trees with two labels, Electron. J. Combin. 9 (2002/03), no. 2, Research paper 19, 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, Unpublished Memorandum, Jan 26 1978

S. Burrill, S. Melczer, and M. Mishna, A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape, arXiv preprint arXiv:1411.6606 [math.CO], 2014-2015.

H. Canary, Aztec diamonds and Baxter permutations, arXiv:math/0309135 [math.CO], 2003-2004.

G. Chatel and V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.

T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.

F. R. K. Chung, R. L. Graham, V. E. Hoggatt Jr. and M. Kleiman, The number of Baxter permutations J. Combin. Theory Ser. A 24 (1978), no. 3, 382-394.

J. 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, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019).

S. Dulucq and O. Guibert, Permutations de Baxter, Séminaire Lotharingien de Combinatoire, B33c (1994), 9 pp.

S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.

S. Dulucq and O. Guibert, Baxter permutations, Discrete Math. 180 (1998), no. 1-3, 143--156. MR1603713 (99c:05004).

Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.

D. C. Fielder and C. O. Alford, An investigation of sequences derived from Hoggatt Sums and Hoggatt Triangles, Application of Fibonacci Numbers, 3 (1990) 77-88. Proceedings of 'The Third Annual Conference on Fibonacci Numbers and Their Applications,' Pisa, Italy, July 25-29, 1988. (Annotated scanned copy)

D. C. Fielder and C. O. Alford, On a conjecture by Hoggatt with extensions to Hoggatt sums and Hoggatt triangles, Fib. Quart., 27 (1989), 160-168.

P. Gawrychowski and P. K. Nicholson, Optimal Encodings for Range Top-k, Selection, and Min-Max, arXiv:1411.6581 [cs.DS], 2014-2015.

P. Gawrychowski and P. K. Nicholson, Optimal Encodings for Range Top-k, Selection, and Min-Max, in Automata, Languages, and Programming, Volume 9134 of the series Lecture Notes in Computer Science, 2015, pp. 593-604.

S. Giraudo, Algebraic and combinatorial structures on Baxter permutations, DMCTS proc. AO, FPSAC 2011 Rykjavik, (2011) 387-398.

S. Giraudo, Algebraic and combinatorial structures on pairs of twin binary trees, arXiv:1204.4776 [math.CO], 2012.

S. Giraudo, Algebraic and combinatorial structures on pairs of twin binary trees, Journal of Algebra, Volume 360, 15 June 2012, Pages 115-157.

O. Guibert and S. Linusson, Doubly alternating Baxter permutations are Catalan, Discrete Math., 217 (2000), 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.

V. E. Hoggatt, Jr., Letter to N. J. A. Sloane, Apr 1977

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.

A. Peder and M. Tombak, Finding the description of structure by counting method: a case study, SOFSEM 2011, LNCS 6543 (2011) 455-466 doi:10.1007/978-3-642-18381-2_38.

Vincent Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, arXiv:1505.07665 [math.CO], 2015.

Jannik Silvanus, Improved Cardinality Bounds for Rectangle Packing Representations, Doctoral Dissertation, University of Bonn (Rheinische Friedrich Wilhelms Universität, Germany 2019).

N. J. A. Sloane, Three vicious walkers of lengths 1, 2, and 3

V. Reiner, D. Stanton, and V. Welker, The Charney-Davis quantity for certain graded posets, Sem. Lothar. Combin. 50 (2003/04), Art. B50c, 13 pp.

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

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. - Richard L. Ollerton, Sep 13 2006

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

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;

# second Maple program:

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]=0; 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 = 0

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) [0] 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

(Sage) [0]+[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

(GAP) Concatenation([0], 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

Cf. A001183, A001185, A046996, A006002, A007318.

Sequence in context: A342290 A342293 A342291 * A130579 A279570 A338748

Adjacent sequences:  A001178 A001179 A001180 * A001182 A001183 A001184

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

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

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Last modified October 5 16:55 EDT 2022. Contains 357259 sequences. (Running on oeis4.)