A214015 -id:A214015 - cp's OEIS frontend

A056986 Number of permutations on {1,...,n} containing any given pattern alpha in the symmetric group S_3.

0, 0, 1, 10, 78, 588, 4611, 38890, 358018, 3612004, 39858014, 478793588, 6226277900, 87175616760, 1307664673155, 20922754530330, 355687298451210, 6402373228089300, 121645098641568810, 2432902001612519580, 51090942147243172980, 1124000727686125116360

Offset: 1

Eric W. Weisstein

This is well-defined because for all patterns alpha in S_3 the number of permutations in S_n avoiding alpha is the same (the Catalan numbers). - Emeric Deutsch, May 05 2008

			a(4) = 10 because, taking, for example, the pattern alpha=321, we have 3214, 3241, 1432, 2431, 3421, 4213, 4132, 4231, 4312 and 4321.

Alois P. Heinz, Table of n, a(n) for n = 1..170
FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern [1,2,3] inside a permutation of length at least 3, The number of occurrences of the pattern [1,3,2] in a permutation, The number of occurrences of the pattern [2,1,3] in a permutation, The number of occurrences of the pattern [2,3,1] in a permutation, The number of occurrences of the pattern [3,1,2] in a permutation
R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, pp. 383-406, 1985.
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A000108, A000142, A138159, A214015, A214152.

A056986:= func< n | Factorial(n) - Catalan(n) >;
[A056986(n): n in [1..30]]; // G. C. Greubel, Oct 06 2024

a:= n-> n! -binomial(2*n, n)/(n+1):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 05 2012

Mathematica
```
Table[n! -CatalanNumber[n], {n,30}]
```

a(n)=n!-binomial(n+n,n+1)/n \\ Charles R Greathouse IV, Jun 10 2011

def A056986(n): return factorial(n) - catalan_number(n)
[A056986(n) for n in range(1,31)] # G. C. Greubel, Oct 06 2024

From Alois P. Heinz, Jul 05 2012: (Start)
a(n) = A214152(n, 3).
a(n) = A000142(n) - A000108(n).
a(n) = A000142(n) - A214015(n, 2). (End)
E.g.f.: 1/(1 - x) - exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)). - Ilya Gutkovskiy, Jan 21 2017

A158005 Numbers of pattern-matching permutations of (1234) for the permutations of {1, 2, ..., n} on n = 4, 5, 6, ... elements.

Original entry on oeis.org

1, 17, 207, 2279, 24553, 268521, 3042210, 36153510, 454208895, 6059942223, 86030083110, 1299647574882, 20865826165777, 355277740280849, 6399391841784282, 121623163346687166, 2432739049821421911, 51089720946192154791, 1123991502048375026337

Offset: 4

Eric W. Weisstein, Mar 11 2009

nonn

Same series for 1243 1432 2134 2143 4123 3214 3412 2341 3421 4321 4312. - R. H. Hardin, Mar 15 2009

Alois P. Heinz, Table of n, a(n) for n = 4..170
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A000142, A005802, A214015, A214152.

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l)
      `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> n! -g(n, 3, []):
seq(a(n), n=4..30);  # Alois P. Heinz, Jul 05 2012
# second Maple program
a:= proc(n) option remember; `if`(n<3, 0, `if`(n=4, 1,
      ((13-11*n-40*n^2+10*n^3+n^4)*a(n-1) -(10*n^2-9*n-31)*(n-1)^2*a(n-2)
       +9*(n-1)^2*(n-2)^2*a(n-3)) / ((n-4)*(n+2)^2)))
    end:
seq(a(n), n=4..30);  # Alois P. Heinz, Sep 26 2012

a[2] = a[3] = 0; a[4] = 1; a[n_] := a[n] = (1/((n-4)*(n+2)^2))* (9*(n-2)^2*a[n-3]*(n-1)^2 - (10*n^2 - 9*n - 31)*a[n-2]*(n-1)^2 + (n^4 + 10*n^3 - 40*n^2 - 11*n + 13)*a[n-1]); Table[a[n], {n, 4, 22}] (* Jean-François Alcover, Oct 22 2012, after Alois P. Heinz *)

a(n) = A214152(n,4) = A000142(n) - A005802(n) = A000142(n) - A214015(n,3). - Alois P. Heinz, Jul 05 2012

More terms from R. H. Hardin, Mar 15 2009
Two more terms from Vladeta Jovovic, Aug 17 2009
Corrected a(19)-a(20) and extended by Alois P. Heinz, Jul 05 2012

A005802 Number of permutations in S_n with longest increasing subsequence of length <= 3 (i.e., 1234-avoiding permutations); vexillary permutations (i.e., 2143-avoiding).

Original entry on oeis.org

1, 1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705, 167078577, 1148208090, 8026793118, 56963722223, 409687815151, 2981863943718, 21937062144834, 162958355218089, 1221225517285209, 9225729232653663, 70209849031116183, 537935616492552297

Offset: 0

N. J. A. Sloane, Simon Plouffe

Also the dimension of SL(3)-invariants in V^n tensor (V^*)^n, where V is the standard 3-dimensional representation of SL(3) and V^* is its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
Also the number of doubly-alternating permutations of length 2n with no four-term increasing subsequence (i.e., 1234-avoiding doubly-alternating permutations). The doubly-alternating permutations (counted by sequence A007999) are those permutations w such that both w and w^(-1) have descent set {2, 4, 6, ...}. - Joel B. Lewis, May 21 2009
Any permutation without an increasing subsequence of length 4 has a decreasing subsequence of length >= n/3, where n is the length of the sequence, by the Erdős-Szekeres theorem. - Charles R Greathouse IV, Sep 26 2012
Also the number of permutations of length n simultaneously avoiding patterns 1324 and 3416725 (or 1324 and 3612745). - Alexander Burstein, Jan 31 2014

Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, pp. 65-82 of "A Century of Advancing Mathematics", ed. S. F. Kennedy et al., MAA Press 2015.
S. Kitaev, Patterns in Permutations and Words, Springer-Verlag, 2011. see p. 399 Table A.7.
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 7.16(e), p. 453.

Alois P. Heinz, Table of n, a(n) for n = 0..1060
Michael Albert and Mireille Bousquet-Mélou, Permutations sortable by two stacks in parallel and quarter plane walks, arXiv:1312.4487 [math.CO], 2014-2015.
Michael Albert and Mireille Bousquet-Mélou, Permutations sortable by two stacks in parallel and quarter plane walks, European Journal of Combinatorics 43 (2015), 131-164.
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. Burstein and J. Pantone, Two examples of unbalanced Wilf-equivalence, arXiv preprint arXiv:1402.3842 [math.CO], 2014.
CombOS - Combinatorial Object Server, Generate pattern-avoiding permutations
Andrew R. Conway and Anthony J. Guttmann, On 1324-avoiding permutations, Advances in Applied Mathematics, Volume 64, March 2015, p. 50-69.
Andrew R. Conway and Anthony J. Guttmann, Counting occurrences of patterns in permutations, arXiv:2306.12682 [math.CO], 2023. See p. 18.
Tom Denton, Algebraic and Affine Pattern Avoidance, arXiv preprint arXiv:1303.3767 [math.CO], 2013.
Eric S. Egge, Defying God: The Stanley-Wilf Conjecture, Stanley-Wilf Limits, and a Two-Generation Explosion of Combinatorics, MAA Focus, August/September 2015, pp. 33-34. [Annotated scanned copy]
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015.
Steven Finch, Pattern-Avoiding Permutations [Cached copy at the Wayback Machine]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]
A. L. L. Gao, S. Kitaev, and P. B. Zhang, On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
O. Guibert and E. Pergola, Enumeration of vexillary involutions which are equal to their mirror/complement, Discrete Math., Vol. 224 (2000), pp. 281-287.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, and Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With A Prescribed Number of "Forbidden" Patterns.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With A Prescribed Number of "Forbidden" Patterns, Advances in Applied Mathematics, Vol. 17, 1996, pp. 381-407.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With a Prescribed Number of "Forbidden" Patterns, arXiv:math/9808080 [math.CO], 1998.
Erik Ouchterlony, Pattern avoiding doubly alternating permutations
Permutation Pattern Avoidance Library (PermPAL), 0123
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Anders Bjorner and Richard P. Stanley, A combinatorial miscellany
R. P. Stanley, Recent Progress in Algebraic Combinatorics, Bull. Amer. Math. Soc., 40 (2003), 55-68.

A column of A047888. See also A224318, A223034, A223905.
Column k=3 of A214015.
A005802, A022558, A061552 are representatives for the three Wilf classes for length-four avoiding permutations (cf. A099952).

a:= n-> 2*add(binomial(2*k,k)*(binomial(n,k))^2*(3*k^2+2*k+1-n-2*k*n)/ (k+1)^2/(k+2)/(n-k+1),k=0..n);
A005802:=rsolve({a(0) = 1, a(1) = 1, (n^2 + 8*n + 16)*a(n + 2) = (10*n^2 + 42*n + 41)*a(n + 1) - (9*n^2 + 18*n + 9)*a(n)},a(n),makeproc): # Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

a[n_] := 2Sum[Binomial[2k, k]Binomial[n, k]^2(3k^2+2k+1-n-2k*n)/((k+1)^2(k+2)(n-k+1)), {k, 0, n}]
(* Second program:*)
a[0] = a[1] = 1; a[n_] := a[n] = ((10*n^2+2*n-3)*a[n-1] + (-9*n^2+18*n-9)* a[n-2])/(n+2)^2; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 20 2017 *)
Table[HypergeometricPFQ[{1/2, -1 - n, -n}, {2, 2}, 4] / (n+1), {n, 0, 25}] (* Vaclav Kotesovec, Jun 07 2021 *)

a(n)=2*sum(k=0,n,binomial(2*k,k)*binomial(n,k)^2*(3*k^2+2*k+1-n-2*k*n)/(k+1)^2/(k+2)/(n-k+1)) \\ Charles R Greathouse IV, Sep 26 2012

a(n) = 2 * Sum_{k=0..n} binomial(2*k, k) * (binomial(n, k))^2 * (3*k^2 + 2*k+1 - n - 2*k*n)/((k+1)^2 * (k+2) * (n-k+1)).
(4*n^2 - 2*n + 1)*(n + 2)^2*(n + 1)^2*a(n) = (44*n^3 - 14*n^2 - 11*n + 8)*n*(n + 1)^2*a(n - 1) - (76*n^4 + 42*n^3 - 49*n^2 - 24*n + 24)*(n - 1)^2*a(n - 2) + 9*(4*n^2 + 6*n + 3)*(n - 1)^2*(n - 2)^2*a(n - 3). - Vladeta Jovovic, Jul 16 2004
a(0) = 1, a(1) = 1, (n^2 + 8*n + 16)*a(n + 2) = (10*n^2 + 42*n + 41) a(n + 1) - (9*n^2 + 18*n + 9) a(n). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
a(n) = ((18*n+45)*A002893(n) - (7+2*n)*A002893(n+1)) / (6*(n+2)^2). - Mark van Hoeij, Jul 02 2010
G.f.: (1+5*x-(1-9*x)^(3/4)*(1-x)^(1/4)*hypergeom([-1/4, 3/4],[1],64*x/((x-1)*(1-9*x)^3)))/(6*x^2). - Mark van Hoeij, Oct 25 2011
a(n) ~ 3^(2*n+9/2)/(16*Pi*n^4). - Vaclav Kotesovec, Jul 29 2013
a(n) = Sum_{k=0..n} binomial(2k,k)*binomial(n+1,k+1)*binomial(n+2,k+1)/((n+1)^2*(n+2)). [Conway and Guttmann, Adv. Appl. Math. 64 (2015) 50]
For n > 0, (n+2)^2*a(n) - n^2*a(n-1) = 4*A086618(n). - Zhi-Wei Sun, Nov 16 2017
a(n) = hypergeom([1/2, -1 - n, -n], [2, 2], 4) / (n+1). - Vaclav Kotesovec, Jun 07 2021

Additional comments from Emeric Deutsch, Dec 06 2000
More terms from Naohiro Nomoto, Jun 18 2001
Edited by Dean Hickerson, Dec 10 2002
More terms from Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

A047889 Number of permutations in S_n with longest increasing subsequence of length <= 4.

Original entry on oeis.org

1, 1, 2, 6, 24, 119, 694, 4582, 33324, 261808, 2190688, 19318688, 178108704, 1705985883, 16891621166, 172188608886, 1801013405436, 19274897768196, 210573149141896, 2343553478425816, 26525044132374656, 304856947930144656

Offset: 0

Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane

Or, number of permutations in S_n that avoid the pattern 12345, - N. J. A. Sloane, Mar 19 2015

Also, the dimension of the space of SL(4)-invariants in V^m ⊗ (V^*)^m, where V is the standard 4-dimensional representation of SL(4) and V^* its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

			G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 119*x^5 + 694*x^6 + 4582*x^7 + ...

Gheorghe Coserea, Table of n, a(n) for n = 0..300
F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015.
Peter J. Forrester and Fei Wei, Higher order linear differential equations for unitary matrix integrals: applications and generalisations, arXiv:2508.20797 [math-ph], 2025.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
Permutation Pattern Avoidance Library (PermPAL), 12345
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Index entries for sequences related to Young tableaux.

A column of A047888.
Cf. A005802, A047890, A052399.
Column k=4 of A214015.
Representatives for the 16 Wilf-equivalence patterns of length 5 are given in A116485, A047889, and A256195-A256208. - N. J. A. Sloane, Mar 19 2015

A:=rsolve({a(0) = 1, a(1) = 1, (n^3 + 16*n^2 + 85*n + 150)*a(n + 2) = (20*n^3 + 182*n^2 + 510*n + 428)*a(n + 1) - (64*n^3 + 256*n^2 + 320*n +128)*a(n)}, a(n), makeproc): # Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

Flatten[{1,RecurrenceTable[{64*(-1+n)^2*n*a[-2+n]-2*(-12 + 11*n + 31*n^2 + 10*n^3)*a[-1+n] + (3+n)^2*(4+n)*a[n]==0,a[1]==1,a[2]==2},a,{n,20}]}] (* Vaclav Kotesovec, Sep 10 2014 *)

{a(n) = my(v); if( n<2, n>=0, v = vector(n+1, k, 1); for(k=2, n, v[k+1] = ((20*k^3 + 62*k^2 + 22*k - 24) * v[k] - 64*k*(k-1)^2 * v[k-1]) / ((k+3)^2 * (k+4))); v[n+1])}; /* Michael Somos, Apr 19 2015 */

a(0)=1, a(1)=1, (n^3 + 16*n^2 + 85*n + 150)*a(n+2) = (20*n^3 + 182*n^2 + 510*n + 428)*a(n+1) - (64*n^3 + 256*n^2 + 320*n + 128)*a(n). - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005
a(n) = (64*(n+1)*(2*n^3 + 21*n^2 + 76*n + 89)*A002895(n) -(8*n^4 + 104*n^3 + 526*n^2 + 1098*n + 776) *A002895(n+1)) / (3*(n+2)^3*(n+3)^3*(n+4)). - Mark van Hoeij, Jun 02 2010
a(n) ~ 3 * 2^(4*n + 9) / (n^(15/2) * Pi^(3/2)). - Vaclav Kotesovec, Sep 10 2014

More terms from Naohiro Nomoto, Mar 01 2002

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

A214152 Number of permutations T(n,k) in S_n containing an increasing subsequence of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 24, 23, 10, 1, 120, 119, 78, 17, 1, 720, 719, 588, 207, 26, 1, 5040, 5039, 4611, 2279, 458, 37, 1, 40320, 40319, 38890, 24553, 6996, 891, 50, 1, 362880, 362879, 358018, 268521, 101072, 18043, 1578, 65, 1, 3628800, 3628799, 3612004, 3042210, 1438112, 337210, 40884, 2603, 82, 1

Offset: 1

Alois P. Heinz, Jul 05 2012

			T(3,2) = 5.  All 3! = 6 permutations of {1,2,3} contain an increasing subsequence of length 2 with the exception of 321.
Triangle T(n,k) begins:
     1;
     2,    1;
     6,    5,    1;
    24,   23,   10,    1;
   120,  119,   78,   17,   1;
   720,  719,  588,  207,  26,  1;
  5040, 5039, 4611, 2279, 458, 37,  1;
  ...

Alois P. Heinz, Rows n = 1..55, flattened
Eric Weisstein's World of Mathematics, Permutation Pattern
Wikipedia, Longest increasing subsequence problem
Wikipedia, Young tableau

Columns k=1-10 give: A000142 (for n>0), A033312, A056986, A158005, A158432, A159139, A159175, A217675, A217676, A217677.
Row sums give: A003316.
T(2n,n) gives A269021.
Diagonal and lower diagonals give: A000012, A002522, A217200, A217193.
Cf. A047874, A214015.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
                 add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
T:= (n, k)-> n! -g(n, k-1, []):
seq(seq(T(n, k), k=1..n), n=1..12);

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]! / Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}] ]; g[n_, i_, l_] := If[n == 0 || i === 1, h[Join[l, Array[1&, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; t[n_, k_] := n! - g[n, k-1, {}]; Table[Table[t[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

T(n,k) = Sum_{i=k..n} A047874(n,i).

T(n,k) = A000142(n) - A214015(n,k-1).

A047890 Number of permutations in S_n with longest increasing subsequence of length <= 5.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 719, 5003, 39429, 344837, 3291590, 33835114, 370531683, 4285711539, 51990339068, 657723056000, 8636422912277, 117241501095189, 1639974912709122, 23570308719710838, 347217077020664880, 5231433025400049936, 80466744544235325387

Offset: 0

Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane

Alois P. Heinz, Table of n, a(n) for n = 0..735
F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513, 2015.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
Permutation Pattern Avoidance Library (PermPAL), 012345
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Index entries for sequences related to Young tableaux.

A column of A047888. Cf. A005802, A052399.

Column k=5 of A214015.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
      +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l)
      `if`(n=0 or i=1, h([l[], 1$n])^2, `if`(i<1, 0,
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)))
    end:
a:= n-> g(n, 5, []):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 10 2012
# second Maple program
a:= proc(n) option remember; `if`(n<6, n!, ((-375+400*n+843*n^2
       +322*n^3+35*n^4)*a(n-1) +225*(n-1)^2*(n-2)^2*a(n-3)
       -(259*n^2+622*n+45)*(n-1)^2*a(n-2))/ ((n+6)^2*(n+4)^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 26 2012

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_, k_] := If[k >= n, n!, g[n, k, {}]]; Table[a[n, 5], {n, 1, 30}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

a(n) ~ 9 * 5^(2*n + 25/2) / (512 * n^12 * Pi^2). - Vaclav Kotesovec, Sep 10 2014

More terms from Naohiro Nomoto, Mar 01 2002

More terms from Alois P. Heinz, Apr 10 2012

A052399 Number of permutations in S_n with longest increasing subsequence of length <= 6.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5039, 40270, 361302, 3587916, 38957991, 457647966, 5763075506, 77182248916, 1091842643475, 16219884281650, 251774983140578, 4066273930979460, 68077194367392864, 1177729684507324152, 20995515989327134152, 384762410996641402384

Offset: 0

N. J. A. Sloane, Mar 13 2000

nonn

Previous name was: Related to Young tableaux of bounded height.

Alois P. Heinz, Table of n, a(n) for n = 0..250
F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015.
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
Index entries for sequences related to Young tableaux.

Cf. A005802, A047889, A047890.

Column k=6 of A214015.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
       +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) option remember;
      `if`(n=0, h(l)^2, `if`(i<1, 0, `if`(i=1, h([l[], 1$n])^2,
       g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
a:= n-> g(n, 6, []):
seq(a(n), n=0..25); # Alois P. Heinz, Apr 10 2012
# second Maple program
a:= proc(n) option remember; `if`(n<7, n!,
      ((56*n^5-9408+11032*n+19028*n^2+7360*n^3+1092*n^4)*a(n-1)
       -4*(196*n^3+1608*n^2+3167*n+444)*(n-1)^2*a(n-2)
       +1152*(2*n+3)*(n-1)^2*(n-2)^2*a(n-3))/ ((n+9)*(n+8)^2*(n+5)^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 26 2012

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; a[n_, k_] := If[k >= n, n!, g[n, k, {}]]; Table[a[n, 6], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)

a(n) ~ 5 * 2^(2*n + 6) * 3^(2*n + 21) / (n^(35/2) * Pi^(5/2)). - Vaclav Kotesovec, Sep 10 2014

More terms from Alois P. Heinz, Apr 10 2012

New name from Vaclav Kotesovec, Sep 10 2014

A047888 Rectangular array of numbers a(n,k) = number of permutations of n things with longest increasing subsequence of length <= k (1 <= k <= oo), read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 2, 1, 1, 14, 6, 2, 1, 1, 42, 23, 6, 2, 1, 1, 132, 103, 24, 6, 2, 1, 1, 429, 513, 119, 24, 6, 2, 1, 1, 1430, 2761, 694, 120, 24, 6, 2, 1, 1, 4862, 15767, 4582, 719, 120, 24, 6, 2, 1, 1, 16796, 94359, 33324, 5003, 720, 120, 24, 6, 2, 1, 1, 58786, 586590

Offset: 1

Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane

Also a(n,k) is the dimension of the space of SL(k)-invariants in V^n tensor (V^*)^n, where V is the standard k-dimensional representation of SL(k) and V^* is its dual. - Alec Mihailovs (alec(AT)mihailovs.com), Aug 14 2005

			Square array a(n,k) begins:
  1,   1,    1,    1,    1,    1, ...
  1,   2,    2,    2,    2,    2, ...
  1,   5,    6,    6,    6,    6, ...
  1,  14,   23,   24,   24,   24, ...
  1,  42,  103,  119,  120,  120, ...
  1, 132,  513,  694,  719,  720, ...

Alois P. Heinz, Antidiagonals n = 1..44, flattened
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53, no. 2, (1990), 257-285.

Rows of the array are partial sums of A047874. Cf. A047887.

Subarray of A214015.

rows = 12; h[l_List] := Module[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]] ; g[n_, i_, l_List] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]]^2, If[i<1, 0, Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; T[n_] := Table[g[n-k, Min[n-k, k], {k}], {k, 1, rows}] // Accumulate; A047888 = Table[T[n], {n, 1, rows}]; Table[A047888[[n-k+1, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 06 2014, after Alois P. Heinz *)

b(n, k) = {
  my(x = 'x + O('x^(2*n)));
  sum(i = 0, n, x^(2*i+k)/(i!*(i+k)!));
};
u(n, k) = {
  my(v = Vec(matdet(matrix(k, k, i, j, b(n, abs(i-j))))));
  return(vector((#v-1)\2, i, v[2*i+1] * i!^2));
};
A(n, k) = {
  my(m = [;]);
  for (i = 1, k, m = concat(m, u(n, i)~));
  return(m);
};
A(6, 6)  \\ Gheorghe Coserea, Feb 02 2016

More terms from Naohiro Nomoto, Mar 01 2002

A267479 Number A(n,k) of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 6, 1, 0, 1, 1, 6, 43, 1, 0, 1, 1, 6, 90, 352, 1, 0, 1, 1, 6, 90, 1879, 3114, 1, 0, 1, 1, 6, 90, 2520, 47024, 29004, 1, 0, 1, 1, 6, 90, 2520, 102011, 1331664, 280221, 1, 0, 1, 1, 6, 90, 2520, 113400, 5176504, 41250519, 2782476, 1, 0

Offset: 0

Alois P. Heinz, Jan 15 2016

			Square array A(n,k) begins:
  1, 1,     1,       1,       1,       1,       1, ...
  0, 1,     1,       1,       1,       1,       1, ...
  0, 1,     6,       6,       6,       6,       6, ...
  0, 1,    43,      90,      90,      90,      90, ...
  0, 1,   352,    1879,    2520,    2520,    2520, ...
  0, 1,  3114,   47024,  102011,  113400,  113400, ...
  0, 1, 29004, 1331664, 5176504, 7235651, 7484400, ...

Alois P. Heinz, Antidiagonals n = 0..30, flattened
Ferenc Balogh, A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length, arXiv:1505.01389, 2015
Shalosh B. Ekhad and Doron Zeilberger, The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ..., n are D-finite for all d and all r, 2014

Columns k=0-4 give: A000007, A000012, A220097, A266734, A266735.
Main diagonal gives A000680.
First lower diagonal gives A267532.
Cf. A214015, A267480.

A(n,k) = Sum_{i=0..k} A267480(n,i).

A047887 Triangle of numbers T(n,k) = number of permutations of n things with longest increasing subsequence of length <=k (1<=k<=n).

Original entry on oeis.org

1, 1, 2, 1, 5, 6, 1, 14, 23, 24, 1, 42, 103, 119, 120, 1, 132, 513, 694, 719, 720, 1, 429, 2761, 4582, 5003, 5039, 5040, 1, 1430, 15767, 33324, 39429, 40270, 40319, 40320, 1, 4862, 94359, 261808, 344837, 361302, 362815, 362879, 362880, 1, 16796

Offset: 1

Eric Rains (rains(AT)caltech.edu), N. J. A. Sloane

			Triangle T(n,k) begins:
  1;
  1,   2;
  1,   5,    6;
  1,  14,   23,   24;
  1,  42,  103,  119,  120;
  1, 132,  513,  694,  719,  720;
  1, 429, 2761, 4582, 5003, 5039, 5040;
  ...

Alois P. Heinz, Rows n = 1..45, flattened
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.

Rows are partial sums of A047874.

Cf. A047888, A214015.

h[l_] := Module[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Array[1 &, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]];
T[n_] := Table[g[n - k, Min[n - k, k], {k}], {k, 1, n}] // Accumulate;
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 24 2016, after Alois P. Heinz *)

More terms from Naohiro Nomoto, Mar 01 2002

cp's OEIS Frontend

A056986 Number of permutations on {1,...,n} containing any given pattern alpha in the symmetric group S_3.

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A158005 Numbers of pattern-matching permutations of (1234) for the permutations of {1, 2, ..., n} on n = 4, 5, 6, ... elements.

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A005802 Number of permutations in S_n with longest increasing subsequence of length <= 3 (i.e., 1234-avoiding permutations); vexillary permutations (i.e., 2143-avoiding).

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A047889 Number of permutations in S_n with longest increasing subsequence of length <= 4.

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A214152 Number of permutations T(n,k) in S_n containing an increasing subsequence of length k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A047890 Number of permutations in S_n with longest increasing subsequence of length <= 5.

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A052399 Number of permutations in S_n with longest increasing subsequence of length <= 6.

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