Miklos Bona - cp's OEIS frontend

User: Miklos Bona

Miklos Bona has authored 8 sequences.

A309563 Cyclic permutations of length n that avoid the patterns 123 and 231.

1, 1, 1, 1, 2, 3, 2, 2, 4, 6, 2, 2, 6, 8, 4, 4, 6, 10, 4, 4, 10, 12, 4, 4, 12, 18, 6, 6, 8, 12, 8, 8, 16, 22, 6, 6, 18, 22, 8, 8, 12, 22, 10, 10, 22, 26, 8, 8, 20, 32, 12, 12, 18, 24, 12, 12, 28, 36, 8, 8, 30, 38, 16, 16, 20, 36, 16, 16, 24, 30, 12, 12, 36, 54

Offset: 1

Miklos Bona, Aug 07 2019

nonn

			a(4)=1, since the only cyclic permutation of length 4 avoiding both 123 and 231 is (4231)=4312.

Miklos Bona, Michael Cory, Cyclic Permutations Avoiding Pairs of Patterns of Length Three, arXiv:1805.05196 [math.CO], 2018

Cf. A000010.

a(2)=1; a(n)=phi(n/2) if n=4k, a(n)=phi((n+2)/4) +phi(n/2) if n=4k+2 > 2, and a(n)=phi((n+1)/2) if n is odd, where phi is the Euler totient function.

A309508 Number of cyclic permutations of length n avoiding the pattern 321.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 24, 66, 178, 512, 1486, 4446, 13468, 41648, 130178, 412670, 1321418, 4274970, 13948966, 45890440, 152061154, 507292698, 1702753462, 5748085332, 19506240462

Offset: 0

Miklos Bona, Aug 05 2019

Comment from F. Chapoton, Sep 14 2021: (Start)
The maps sending a permutation to its inverse or to its reverse-complement define two commuting involutions on these sets of permutations.
The next terms in the sequence could be 41648, 130178, though these are counting Dyck words such that an associated permutation is cyclic, related but not obviously equivalent combinatorial objects. (End)

			For n=3, there are two such permutations, 231 and 312.
The a(4) = 4 permutations are: 2341, 2413, 3142, 4123.
The a(5) = 10 permutations are: 23451, 23514, 24153, 25134, 31452, 31524, 34512, 41253, 45123, 51234.

Kassie Archer, Christina Graves, and Robert Laudone, Binary operations on pattern-avoiding cycles, arXiv:2505.04456 [math.CO], 2025.
Miklos Bona and Michael Cory, Cyclic Permutations Avoiding Pairs of Patterns of Length Three, arXiv:1805.05196 [math.CO], 2018.

Cf. A000108 (number of permutations avoiding 321).

Cf. A000957, A309504, A309505, A309506.

\\ See PARI link in A309504 for program code.
for(n=0, 16, print1(E321(n), ", ")) \\ Andrew Howroyd, Nov 20 2024

a(0)=1 prepended and a(13)-a(24) from Andrew Howroyd, Nov 17 2024

A309506 Number of cyclic permutations of length n avoiding the pattern 231 (equivalently, 312).

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 12, 30, 86, 253, 748, 2274, 7152, 22890, 74189, 243342, 808599, 2716549, 9213420, 31498358, 108483093, 376145636, 1312463081, 4605569378, 16245866825

Offset: 0

Miklos Bona, Aug 05 2019

			For n=4, there are two such permutations, 4123 and 4312.
The a(5) = 5 permutations are 51234, 51423, 53124, 54132, 54213.

Kassie Archer, Christina Graves, and Robert Laudone, Binary operations on pattern-avoiding cycles, arXiv:2505.04456 [math.CO], 2025. See p. 23.
Miklos Bona and Michael Cory, Cyclic Permutations Avoiding Pairs of Patterns of Length Three, arXiv:1805.05196 [math.CO], 2018.

Cf. A000108 (number of permutations avoiding 231).

Cf. A309504, A309505, A309508.

\\ See PARI link in A309504 for program code.
for(n=0, 16, print1(E231(n), ", ")) \\ Andrew Howroyd, Nov 20 2024

a(1)=1 (confirmed by author) inserted by Alexander Burstein, Jul 20 2020

a(0)=1 prepended and a(13)-a(24) from Andrew Howroyd, Nov 20 2024

A309505 Number of cyclic permutations of length n that avoid the pattern 132 (equivalently, 213).

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 24, 68, 182, 544, 1574, 4888, 14864, 47610, 149964, 491802, 1592198, 5318936, 17593170, 59679516, 200805614, 689988886, 2354489616, 8178944510, 28240716098

Offset: 0

Miklos Bona, Aug 05 2019

			For n=3, there are two such permutations, 231 and 312.
The a(4) = 4 permutations are: 2341, 3421, 4123, 4312.
The a(5) = 10 permutations are: 23451, 34251, 34512, 43521, 45123, 45231, 51234, 53124, 53412, 54213.

Kassie Archer, Christina Graves, and Robert Laudone, Binary operations on pattern-avoiding cycles, arXiv:2505.04456 [math.CO], 2025.
Miklos Bona and Michael Cory, Cyclic Permutations Avoiding Pairs of Patterns of Length Three, arXiv:1805.05196 [math.CO], 2018.
Brice Huang, An Upper Bound on the Number of (132,213)-Avoiding Cyclic Permutations, arXiv:1808.08462 [math.CO], 2018-2019.

Cf. A000108 (number of permutations avoiding 132).

Cf. A309504, A309506, A309508.

\\ See PARI link in A309504 for program code.
for(n=1, 16, print1(E213(n), ", ")) \\ Andrew Howroyd, Nov 20 2024

a(0)=1 prepended and a(13)-a(24) from Andrew Howroyd, Nov 20 2024

A309504 Number of cyclic permutations of length n avoiding the pattern 123.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 24, 68, 188, 586, 1722, 5492, 16924, 55582, 177278, 594460, 1944980, 6628384, 22132112, 76421498, 259359036, 905416294, 3114033930, 10971347070, 38157201530

Offset: 0

Miklos Bona, Aug 05 2019

			For n=3, there are two such permutations, 231 and 312.
The a(4) = 4 permutations are: 2413, 3142, 3421, 4312.
The a(5) = 10 permutations are: 25413, 35214, 35421, 41532, 43152, 43521, 45231, 53412, 54132, 54213.

Kassie Archer, Christina Graves, and Robert Laudone, Binary operations on pattern-avoiding cycles, arXiv:2505.04456 [math.CO], 2025. See p. 23.
Miklos Bona and Michael Cory, Cyclic Permutations Avoiding Pairs of Patterns of Length Three, arXiv:1805.05196 [math.CO], 2018.
Andrew Howroyd, PARI Program, Nov 2024.

Cf. A000108 (number of permutations avoiding 123).

Cf. A309505, A309506, A309508.

\\ See Links for program code.
for(n=0, 16, print1(E123(n), ", ")) \\ Andrew Howroyd, Nov 20 2024

a(0)=1 prepended and a(13)-a(24) from Andrew Howroyd, Nov 17 2024

A032351 Number of permutations of length n which avoid the patterns 2143, 1324 (smooth permutations); or avoid the patterns 1342, 2431; etc.

Original entry on oeis.org

1, 1, 2, 6, 22, 88, 366, 1552, 6652, 28696, 124310, 540040, 2350820, 10248248, 44725516, 195354368, 853829272, 3733693872, 16333556838, 71476391800, 312865382004, 1369760107576, 5998008630244, 26268304208032, 115055864102504, 503997820344464, 2207927106851580, 9673223726469136, 42382192892577128, 185702341264971696

Offset: 0

Miklos Bona

			1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 88*x^5 + 366*x^6 + 1552*x^7 + ...

S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.47.
R. P. Stanley, Catalan Numbers, Cambridge, 2015, p. 133.

H. Abe and S. Billey, Consequences of the Lakshmibai-Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry, 2014.
George Balla, Ghislain Fourier, and Kunda Kambaso, PBW filtration and monomial bases for Demazure modules in types A and C, arXiv:2205.01747 [math.RT], 2022.
C. Bean, M. Tannock and H. Ulfarsson, Pattern avoiding permutations and independent sets in graphs, arXiv:1512.08155 [math.CO], 2015. See Eq. (2).
Christian Bean, Finding structure in permutation sets, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018.
Miklos Bona, The permutation classes equinumerous to the smooth class, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.
M. Bousquet-Mélou and S. Butler, Forest-like permutations, arXiv:math/0603617 [math.CO], 2006.
Rocco Chirivì, Xin Fang, and Ghislain Fourier, Degenerate Schubert varieties in type A, Transformation Groups (2020).
Darla Kremer and Wai Chee Shiu, Finite transition matrices for permutations avoiding pairs of length four patterns, Discrete Math. 268 (2003), 171-183. MR1983276 (2004b:05006). See Table 1.
E. Rowland and R. Yassawi, Automatic congruences for diagonals of rational functions, arXiv preprint arXiv:1310.8635 [math.NT], 2013-2014.
Wikipedia, Permutation classes avoiding two patterns of length 4.
A. Woo and A. Yong, When is a Schubert variety Gorenstein?, arXiv:math/0409490 [math.AG], 2004.
A. Woo and A. Yong, When is a Schubert variety Gorenstein?, Advances in Mathematics, Volume 207, Issue 1, 1 December 2006, Pages 205-220.

Cf. A053617.

t1:=(1-5*x+3*x^2+x^2*sqrt(1-4*x))/(1-6*x+8*x^2-4*x^3);
series(t1,x,40);
seriestolist(%); # N. J. A. Sloane, Nov 09 2016

Table[(Sum[(m+3)*(Sum[Sum[2^j*Binomial[j+k, k]*Binomial[m-j, 2*k+1], {j, 0, m-2*k-1}], {k, 0, m/2}]) * Binomial[2*n-m-2, n], {m, 0, n-2}] + Binomial[2*n, n])/(n+1),{n,0,20}] (* Vaclav Kotesovec, Sep 19 2014, after Vladimir Kruchinin *)

a(n):=(sum((m+3)*(sum(sum(2^(j)*binomial(j+k,k)*binomial(m-j,2*k+1),j,0,m-2*k-1),k,0,m/2))*binomial(2*n-m-2,n),m,0,n-2)+binomial(2*n,n))/(n+1); /* Vladimir Kruchinin, Sep 19 2014 */

x='x+O('x^44) /* that many terms */
gf=(1-5*x+3*x^2+x^2*sqrt(1-4*x))/(1-6*x+8*x^2-4*x^3);
Vec(gf) /* show terms */ /* Joerg Arndt, Apr 20 2011 */

G.f.: (1-5*x+3*x^2+x^2*sqrt(1-4*x))/(1-6*x+8*x^2-4*x^3).
G.f.: 1 / (1 - x / (1 - x / (1 - 2*x / (1 - x / (1 - x / (1 - x / (1 - x / ...))))))). - Michael Somos, Apr 18 2012
From Gary W. Adamson, Jul 11 2011: (Start)
a(n) = upper left term in n-th power of the following infinite square production matrix:
  1, 1, 0, 0, 0, 0, ...
  1, 2, 1, 0, 0, 0, ...
  1, 3, 1, 1, 0, 0, ...
  1, 4, 1, 1, 1, 0, ...
  1, 5, 1, 1, 1, 1, ...
  ...
(End)
HANKEL transform is A011782. HANKEL transform of a(n+1) is A011782(n+1). INVERT transform of A026671 with 1 prepended. - Michael Somos, Apr 18 2012
Recurrence: (n-2)*a(n) = 2*(5*n-13)*a(n-1) - 4*(8*n-25)*a(n-2) + 12*(3*n-10)*a(n-3) - 8*(2*n-7)*a(n-4). - Vaclav Kotesovec, Aug 24 2014
a(n) ~ 1/11 * (1 - 5*r + 3*r^2 + r^2*sqrt(1-4*r)) *(25 - 44*r + 24*r^2) / r^n, where r = 1/6*(4 - 2/(-17 + 3*sqrt(33))^(1/3) + (-17 + 3*sqrt(33))^(1/3)) = 0.228155493653961819214572... is the root of the equation -1 + 6*r - 8*r^2 + 4*r^3 = 0. - Vaclav Kotesovec, Aug 24 2014
a(n) = (Sum_{m=0..n-2} (m+3)*(Sum_{k=0..m/2} Sum_{j=0..m-2*k-1} 2^j * binomial(j+k, k) * binomial(m-j, 2*k+1)) * binomial(2*n-m-2,n) + binomial(2*n,n))/(n+1). - Vladimir Kruchinin, Sep 19 2014

More terms from Erich Friedman

A026671 Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).

Original entry on oeis.org

1, 3, 11, 43, 173, 707, 2917, 12111, 50503, 211263, 885831, 3720995, 15652239, 65913927, 277822147, 1171853635, 4945846997, 20884526283, 88224662549, 372827899079, 1576001732485, 6663706588179, 28181895551161, 119208323665543, 504329070986033, 2133944799315027

Offset: 0

Clark Kimberling, Miklos Bona

1, 1, 3, 11, 43, 173, ... is the unique sequence for which both the Hankel transform of the sequence itself and the Hankel transform of its left shift are the powers of 2 (A000079). For example, det[{{1, 1, 3}, {1, 3, 11}, {3, 11, 43}}] = det[{{1, 3, 11}, {3, 11, 43}, {11, 43, 173}}] = 4. - David Callan, Mar 30 2007
From Paul Barry, Jan 25 2009: (Start)
a(n) is the image of F(2n+2) under the Catalan matrix (1,xc(x)) where c(x) is the g.f. of A000108.
The sequence 1,1,3,... is the image of A001519 under (1,xc(x)). This sequence has g.f. given by 1/(1-x-2x^2/(1-3x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction). (End)
Binomial transform of A111961. - Philippe Deléham, Feb 11 2009
From Paul Barry, Nov 03 2010: (Start)
The sequence 1,1,3,... has g.f. 1/(1-x/sqrt(1-4x)), INVERT transform of A000984.
It is an eigensequence of the sequence array for A000984. (End)

L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Jean-Christophe Aval, Adrien Boussicault and Sandrine Dasse-Hartaut, The tree structure in staircase tableaux, arXiv:1109.4907 [math.CO], 2011-2013.
Cyril Banderier, Markus Kuba, and Michael Wallner, Analytic Combinatorics of Composition schemes and phase transitions with mixed Poisson distributions, arXiv:2103.03751 [math.PR], 2021.
Miklos Bona, The permutation classes equinumerous to the smooth class, Electron. J. Combin., 5 (1998), no. 1, Research Paper 31, 12 pp.
David Callan and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 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.
Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Huyile Liang, Jeffrey Remmel and Sainan Zheng, Stieltjes moment sequences of polynomials, arXiv:1710.05795 [math.CO], 2017, see page 16.

a(n) = T(2n-1, n-1), T given by A026736.
a(n) = T(2n, n), T given by A026670.
a(n) = T(2n+1, n+1), T given by A026725.
Row sums of triangle A054335.
Cf. A026781, A001076, A176280.

a:=[3,11,43];; for n in [4..30] do a[n]:=(2*(4*n-3)*a[n-1] - 3*(5*n-8)*a[n-2] - 2*(2*n-3)*a[n-3])/n; od; Concatenation([1], a); # G. C. Greubel, Jul 16 2019

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/(Sqrt(1-4*x)-x) )); // G. C. Greubel, Jul 16 2019

Table[SeriesCoefficient[1/(Sqrt[1-4*x]-x),{x,0,n}],{n,0,30}] (* Vaclav Kotesovec, Oct 08 2012 *)

{a(n)= if(n<0, 0, polcoeff( 1/(sqrt(1 -4*x +x*O(x^n)) -x), n))} /* Michael Somos, Apr 20 2007 */

my(x='x+O('x^66)); Vec( 1/(sqrt(1-4*x)-x) ) \\ Joerg Arndt, May 04 2013

(1/(sqrt(1-4*x)-x)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jul 16 2019

From Wolfdieter Lang, Mar 21 2000: (Start)
G.f.: 1/(sqrt(1-4*x)-x).
a(n) = Sum_{i=1..n} a(i-1)*binomial(2*(n-i), n-i) + binomial(2*n, n), n >= 1, a(0)=1. (End)
G.f.: 1/(1 -x -2*x*c(x)) where c(x) = g.f. for Catalan numbers A000108. - Michael Somos, Apr 20 2007
From Paul Barry, Jan 25 2009: (Start)
G.f.: 1/(1 - 3xc(x) + x^2*c(x)^2);
G.f.: 1/(1-3x-2x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-... (continued fraction).
a(0) = 1, a(n) = Sum_{k=0..n} (k/(2n-k))*C(2n-k,n-k)*F(2k+2). (End)
a(n) = Sum_{k=0..n} A039599(n,k) * A000045(k+2). - Philippe Deléham, Feb 11 2009
From Paul Barry, Feb 08 2009: (Start)
G.f.: 1/(1-x/(1-2x/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction);
G.f. of 1,1,3,... is 1/(1-x-2x/(1-x/(1-x/(1-x/(1-... (continued fraction). (End)
From Gary W. Adamson, Jul 14 2011: (Start)
a(n) = the upper left term in M^n, M = the infinite square production matrix:
  3, 2, 0, 0, 0, 0, ...
  1, 1, 1, 0, 0, 0, ...
  1, 1, 1, 1, 0, 0, ...
  1, 1, 1, 1, 1, 0, ...
  1, 1, 1, 1, 1, 1, ...
  ... (End)
From Vaclav Kotesovec, Oct 08 2012: (Start)
D-finite with recurrence: n*a(n) = 2*(4*n-3)*a(n-1) - 3*(5*n-8)*a(n-2) - 2*(2*n-3)*a(n-3).
a(n) ~ (2+sqrt(5))^n/sqrt(5). (End)
a(n) = Sum_{k=0..n+1} 4^(n+1-k) * binomial(n-k/2,n+1-k). - Seiichi Manyama, Mar 30 2025
From Peter Luschny, Mar 30 2025: (Start)
a(n) = 4^n*(binomial(n-1/2, n)*hypergeom([1, (1-n)/2, -n/2], [1/2, 1/2-n], -1/4) + hypergeom([(1-n)/2, 1-n/2], [1-n], -1/4)/4) for n > 0.
a(n) = A001076(n) + A176280(n). (End)

A022558 Number of permutations of length n avoiding the pattern 1342.

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 512, 2740, 15485, 91245, 555662, 3475090, 22214707, 144640291, 956560748, 6411521056, 43478151737, 297864793993, 2059159989914, 14350039389022, 100726680316559, 711630547589023, 5057282786190872, 36132861123763276, 259423620328055093

Offset: 0

Miklos Bona

Differs from A117156 which counts permutations avoiding the *consecutive* pattern 1342. - Ray Chandler, Dec 06 2011
Also, number of permutation of length n avoiding the pattern 3142 (see Stankova (1994) below). - Alexander Burstein, Aug 09 2013
Conjecture: a(n) is the number of permutations of length n simultaneously avoiding patterns 2143 and 415263. - Alexander Burstein, Mar 21 2019

			a(4) = 23 because obviously all permutations of length 4 with the exception of 1342 avoid 1342.

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 768, Th. 12.1.14.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.48.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Miklos Bona, Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps, arXiv:math/9702223 [math.CO], 1997.
Miklos Bona, Exact enumeration of 1342-avoiding permutations; A close link with labeled trees and planar maps, J. Combinatorial Theory, A80 (1997), 257-272.
Alexander Burstein and Jay Pantone, Two examples of unbalanced Wilf-equivalence, J. Combin. 6 (2015), no. 1-2, 55-67.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, and Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
A. R. Conway and A. J. Guttmann, On 1324-avoiding permutations, Adv. Appl. Math. 64 (2015), 50-69.
A. L. L. Gao, S. Kitaev, and P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
C. Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint arXiv:1410.2657 [math.CO], 2014.
Hsien-Kuei Hwang, Mihyun Kang, and Guan-Huei Duh, Asymptotic Expansions for Sub-Critical Lagrangean Forms, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2018.
W. Mlotkowski, K. A. Penson, A Fuss-type family of positive definite sequences, arXiv:1507.07312 (2015), eq. (36).
Z. E. Stankova, Forbidden subsequences, Discrete Math., 132 (1994), no. 1-3, 291-316.
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001. See Fig. 11.

Essentially the same as A004040.
Cf. A117158.
A005802, A022558, A061552 are representatives for the three Wilf classes for length-four avoiding permutations (cf. A099952).

a := proc (n) options operator, arrow: (1/2)*(-1)^(n-1)*(7*n^2-3*n-2)+3*(sum((-1)^(n-i)*2^(i+1)*factorial(2*i-4)*binomial(n-i+2, 2)/(factorial(i)*factorial(i-2)), i = 2 .. n)) end proc: seq(a(n), n = 0 .. 30); # Emeric Deutsch, Oct 15 2014

Table[SeriesCoefficient[32*x/(1+20*x-8*x^2-(1-8*x)^(3/2)),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 07 2012 *)
Table[1/2*(-1)^(n-1) * (-2-3*n+7*n^2) + 1/4*(-1)^n * (1+n) * (-2-13*n+(n+2) * Hypergeometric2F1[-3/2,-n,-2-n,-8]),{n,0,20}] (* Vaclav Kotesovec, Aug 24 2014 *)

x='x+O('x^66); Vec( 32*x/(1+20*x-8*x^2-(1-8*x)^(3/2)) ) \\ Joerg Arndt, May 04 2013

a(n) = (7*n^2-3*n-2)/2 * (-1)^(n-1) + 3*Sum_{i=2..n} 2^(i+1) * (2*i-4)!/(i!*(i-2)!) * binomial(n-i+2, 2) * (-1)^(n-i).
G.f.: 32*x/(1 + 20*x - 8*x^2 - (1 - 8*x)^(3/2)). - Emeric Deutsch, Mar 13 2004
Recurrence: n*a(n) = (7*n-22)*a(n-1) + 4*(2*n-1)*a(n-2). - Vaclav Kotesovec, Oct 07 2012
a(n) ~ 2^(3*n+6)/(243*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Oct 07 2012

Minor edits by Vaclav Kotesovec, Aug 24 2014

cp's OEIS Frontend

User: Miklos Bona

A309563 Cyclic permutations of length n that avoid the patterns 123 and 231.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A309508 Number of cyclic permutations of length n avoiding the pattern 321.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A309506 Number of cyclic permutations of length n avoiding the pattern 231 (equivalently, 312).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A309505 Number of cyclic permutations of length n that avoid the pattern 132 (equivalently, 213).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A309504 Number of cyclic permutations of length n avoiding the pattern 123.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions

A032351 Number of permutations of length n which avoid the patterns 2143, 1324 (smooth permutations); or avoid the patterns 1342, 2431; etc.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

PARI

Formula

Extensions

A026671 Number of lattice paths from (0,0) to (n,n) with steps (0,1), (1,0) and, when on the diagonal, (1,1).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

PARI