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A032351 Number of permutations of length n which avoid the patterns 2143, 1324 (smooth permutations); or avoid the patterns 1342, 2431; etc. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
REFERENCES
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.
LINKS
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.
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.
FORMULA
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
EXAMPLE
1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 88*x^5 + 366*x^6 + 1552*x^7 + ...
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) 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 */
(Maxima)
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 */
CROSSREFS
Cf. A053617.
Sequence in context: A165535 A319028 A165536 * A165537 A165538 A165539
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Erich Friedman
STATUS
approved

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Last modified March 18 22:22 EDT 2024. Contains 370951 sequences. (Running on oeis4.)