A214015 -id:A214015 - cp's OEIS frontend

A217057 Number of permutations in S_n containing exactly one increasing subsequence of length 4.

0, 0, 0, 0, 1, 12, 102, 770, 5545, 39220, 276144, 1948212, 13817680, 98679990, 710108396, 5150076076, 37641647410, 277202062666, 2056218941678, 15358296210724, 115469557503753, 873561194459596, 6647760790457218, 50871527629923754, 391345137795371013

Offset: 0

Alois P. Heinz, Sep 25 2012

nonn

			a(4) = 1: 1234.
a(5) = 12: 12453, 12534, 13425, 13452, 14235, 15234, 23145, 23415, 23451, 31245, 41235, 51234.

Brian Nakamura and Doron Zeilberger, Table of n, a(n) for n = 0..70
Andrew R. Conway and Anthony J. Guttmann, Counting occurrences of patterns in permutations, arXiv:2306.12682 [math.CO], 2023. See p. 16.
Brian Nakamura and Doron Zeilberger, Using Noonan-Zeilberger Functional Equations to enumerate (in Polynomial Time!) Generalized Wilf classes; Local copy, pdf file only, no active links
Brian Nakamura and Doron Zeilberger, Using Noonan-Zeilberger Functional Equations to enumerate (in Polynomial Time!) Generalized Wilf classes, arXiv preprint arXiv:1209.2353, 2012.
Wikipedia, Enumerations of specific permutation classes
Wikipedia, Subsequence

Cf. A005802, A117158, A158005, A214015, A214152.

# programs can be obtained from the Nakamura & Zeilberger link.

A269042 Number of permutations of [2n] avoiding the pattern 12...n.

Original entry on oeis.org

0, 0, 1, 132, 15767, 2190688, 370531683, 77182248916, 19835792076675, 6266271456118776, 2413632612087046844, 1120958514818713738544, 619918692943471064695593, 403190647991638511052901232, 304867528413299672718870216538, 265248225675908889875489731636920

Offset: 0

Alois P. Heinz, Feb 18 2016

nonn

			a(2) = 1: 4321.
a(3) = 132: 165432, 216543, 261543, 265143, 265413, 265431, 316542, ..., 653412, 653421, 654132, 654213, 654231, 654312, 654321.

Alois P. Heinz, Table of n, a(n) for n = 0..30

Cf. A010050, A214015, A267532, A269021.

h:= proc(l) (n-> 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))(nops(l))
    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))):
a:= n-> `if`(n=0, 0, g(2*n, n-1, [])):
seq(a(n), n=0..15);

h[l_] := Function[n, 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}]][Length[l]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Table[1, {n}]]]^2, If[i < 1, 0, Sum[g[n - i*j, i-1, Join[l, Table[i, {j}]]], {j, 0, n/i}]]];
a[n_] := If[n == 0, 0, g[2n, n-1, {}]];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Apr 01 2017, translated from Maple *)

a(n) = (2n)! - A269021(n).
a(n) = A214015(2n,n-1) for n>0.
a(n) ~ (2*n)!. - Vaclav Kotesovec, Mar 26 2016

A072131 T_7(n) in the notation of Bergeron et al., u_k(n) in the notation of Gessel: Related to Young tableaux of bounded height.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40319, 362815, 3626197, 39832877, 476591309, 6162155981, 85494566892, 1264755621000, 19835792076675, 328115505900675, 5698062006852574, 103455252673577866, 1956590161853191160, 38418713005615268760, 780931481835878011620

Offset: 1

Jesse Carlsson (j.carlsson(AT)physics.unimelb.edu.au), Jun 25 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..200
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.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

Cf. A052399 for T_6(n), A047890 for T_5(n), A047889 for T_4(n).

Column k=7 of A214015.

a:= proc(n) option remember; `if`(n<8, n!, ((-343035+429858*n
       +238440*n^3+38958*n^4+634756*n^2+2940*n^5+84*n^6)*a(n-1)
       -(1974*n^4+36336*n^3+213240*n^2+407840*n+82425)*(n-1)^2*a(n-2)
       +2*(49875+42646*n+6458*n^2)*(n-1)^2*(n-2)^2*a(n-3)
       -11025*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4))/ ((n+6)^2*(n+10)^2*(n+12)^2))
    end:
seq (a(n), n=1..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_] := If[n <= 7, n!, g[n, 7, {}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz (A214015) *)

a(n) ~ 6075 * 7^(2*n + 49/2) / (32768 * n^24 * Pi^3). - Vaclav Kotesovec, Sep 10 2014

Typo in title corrected by Joel B. Lewis, Jul 16 2009

A072132 T_8(n) in the notation of Bergeron et al., u_k(n) in the notation of Gessel: Related to Young tableaux of bounded height.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 362879, 3628718, 39912738, 478842196, 6221523082, 87002638276, 1302313974900, 20763508263000, 351019617373500, 6266271456118776, 117671982989344680, 2316256222907194304, 47635421509263043024, 1020455890785584587168

Offset: 1

Jesse Carlsson (j.carlsson(AT)physics.unimelb.edu.au), Jun 25 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..586
F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
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.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

Cf. A052399 for T_6(n), A047890 for T_5(n), A047889 for T_4(n).

Column k=8 of A214015.

a:= proc(n) option remember; `if`(n<4, n!,
      (-147456*(n+4)*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4)
      +128*(33876+30709*n+6687*n^2+410*n^3)*(n-1)^2*(n-2)^2*a(n-3)
      -4*(1092*n^5+37140*n^4+455667*n^3+2387171*n^2+4649270*n+1206000)*
      (n-1)^2*a(n-2) +(-17075520+(22488312+(29223280+(10509820+(1764252+
      (154164+(6804+120*n)*n)*n)*n)*n)*n)*n)*a(n-1))/
      ((n+16)*(n+7)^2*(n+15)^2*(n+12)^2))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 28 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_] := If[n <= 8, n!, g[n, 8, {}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz (A214015) *)

a(n) ~ 1913625 * 2^(6*n + 77) / (n^(63/2) * Pi^(7/2)). - Vaclav Kotesovec, Sep 10 2014

A072133 T_9(n) in the notation of Bergeron et al., u_k(n) in the notation of Gessel: Related to Young tableaux of bounded height.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628799, 39916699, 478995537, 6226736369, 87166698628, 1307240982000, 20907446718225, 355162464899601, 6384776070987990, 121061600999380138, 2413632612087046844, 50453964720806671644, 1102844526263334763556

Offset: 0

Jesse Carlsson (j.carlsson(AT)physics.unimelb.edu.au), Jun 25 2002

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..561
F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
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.
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

Cf. A052399 for T_6(n), A047890 for T_5(n), A047889 for T_4(n).

Column k=9 of A214015.

a:= proc(n) option remember;
      `if`(n<5, n!, ((-1110790863+(1520978576+(1772290401+(607308786+
       (101671498+(9464664+(500874+(14124+165*n)*n)*n)*n)*n)*n)*n)*n)*a(n-1)
       -(1129886062*n+559908333*n^2+111239576*n^3+10655238*n^4+8778*n^6
       +491700*n^5 +353895381)*(n-1)^2*a(n-2) +(258011271+234066216*n
       +58221266*n^2+5463876*n^3 +172810*n^4)*(n-1)^2*(n-2)^2*a(n-3)
       -9*(4070430+1504292*n+117469*n^2)* (n-1)^2*(n-2)^2*(n-3)^2*a(n-4)
       +893025*(n-1)^2*(n-2)^2*(n-3)^2*(n-4)^2*a(n-5)) /
       ((n+20)^2*(n+8)^2*(n+18)^2*(n+14)^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 10 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_] := If[n <= 9, n!, g[n, 9, {}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz (A214015) *)

a(n) ~ 30625 * 3^(4*n + 90) / (2097152 * n^40 * Pi^4). - Vaclav Kotesovec, Sep 10 2014

a(0)=1 prepended by Alois P. Heinz, Feb 09 2017

A158432 Number of permutations of 1..n containing the relative rank sequence { 45312 } at any spacing.

Original entry on oeis.org

1, 26, 458, 6996, 101072, 1438112, 20598112, 300892896, 4521034917, 70286670034, 1135485759114, 19121776482564, 336412530327804, 6191800556586104, 119301546930406184, 2406376964044265344, 50786085223779295344, 1120447461653440780128, 25810064637612342838624

Offset: 5

R. H. Hardin, Mar 18 2009

nonn

Same series for 54321 12345 45321 21345 12354 54312 34521 32145 12543 54123 23451 43215 15432 51234 21354 34512 32154 21543 45123.

Alois P. Heinz, Table of n, a(n) for n = 5..200
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A000142, A047889, A056986, A158005, 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:= (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))):
a:= n-> n! -g(n, 4, []):
seq(a(n), n=5..25);  # Alois P. Heinz, Jul 05 2012
# second Maple program
a:= proc(n) option remember; `if`(n<5, 0, `if`(n=5, 1,
     ((132-142*n-301*n^2-35*n^3+25*n^4+n^5)*a(n-1)
     -2*(10*n^3+33*n^2-181*n-2)*(n-1)^2*a(n-2)
     +64*(n-2)^2*(n-1)^3*a(n-3))/ ((n+4)*(n-5)*(n+3)^2)))
    end:
seq(a(n), n=5..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_] := n! - g[n, 4, {}];
Table[a[n], {n, 5, 25}] (* Jean-François Alcover, Jun 19 2018, after Alois P. Heinz's first program *)

a(n) = A214152(n,5) = A000142(n)-A047889(n) = A000142(n)-A214015(n,4).

Extended beyond a(16) by Alois P. Heinz, Jul 05 2012

A159139 Number of permutations of 1..n containing the relative rank sequence { 213465 } at any spacing.

Original entry on oeis.org

1, 37, 891, 18043, 337210, 6081686, 108469917, 1941309261, 35187952132, 649951312000, 12286366975723, 238445927000811, 4762398793018878, 98074791689121162, 2085684931155975120, 45859509146309390064, 1043533983233372354613, 24590543663448304800169

Offset: 6

R. H. Hardin, Apr 05 2009

nonn

Same series for 654321 123456 564321 213456 123465 654312 456321 321456 123654 654123 345621 432156 126543 651234 564312 456312 321465 213654 564123 345612 432165 216543 561234 234561 543216 165432 612345 456123 321654.

Alois P. Heinz, Table of n, a(n) for n = 6..200
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A000142, A047890, A056986, A158005, A158423, 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, 5, []):
seq(a(n), n=6..30);  # Alois P. Heinz, Jul 05 2012
# second Maple program
a:= proc(n) option remember; `if`(n<6, 0, `if`(n=6, 1,
     ((2475-4819*n^2-2985*n+175*n^4-1021*n^3+n^6+49*n^5)*a(n-1)
     -(35*n^4+441*n^3-845*n^2-4147*n-489)*(n-1)^2*a(n-2)
     +(-1668+329*n+259*n^2)*(n-1)^2*(n-2)^2*a(n-3)
     -225*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4))/ ((n-6)*(n+6)^2*(n+4)^2)))
    end:
seq(a(n), n=6..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_] := n! - g[n, 5, {}];
Table[a[n], {n, 6, 30}] (* Jean-François Alcover, Jun 19 2018, from first Maple program *)

a(n) = A214152(n,6) = A000142(n)-A047890(n) = A000142(n)-A214015(n,5). - Alois P. Heinz, Jul 05 2012

More terms from Alois P. Heinz, Jul 05 2012

A159175 Number of permutations of 1..n containing the relative rank sequence { 1234567 } at any spacing.

Original entry on oeis.org

1, 50, 1578, 40884, 958809, 21353634, 463945294, 9996042284, 215831724525, 4702905606350, 103912444955422, 2336099774748540, 53567906041439136, 1255172323669315848, 30095426182382305848, 739238316780966277616, 18619024923770934306358, 481234428294016650524172

Offset: 7

R. H. Hardin Apr 05 2009

nonn

Same series (among rank sequences with inversion = reversal) for 3214765 2134576.

Alois P. Heinz, Table of n, a(n) for n = 7..200
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A000142, A052399, A056986, A158005, A158423, 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:= (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))):
a:= n-> n! -g(n, 6, []):
seq(a(n), n=7..25);  # Alois P. Heinz, Jul 05 2012
# second Maple program
a:= proc(n) option remember; `if`(n<7, 0, `if`(n=7, 1, ((-93464*n+1072*n^4
      +72128-125284*n^2+84*n^6+994*n^5-30491*n^3+n^7) *a(n-1)
      -4*(14*n^5+399*n^4+1124*n^3-7354*n^2-23983*n-5042)*(n-1)^2 *a(n-2)
      +4*(-7359-2629*n+1596*n^2+196*n^3)*(n-1)^2*(n-2)^2 *a(n-3)
      -1152*(1+2*n)*(n-1)^2*(n-2)^2*(n-3)^2 *a(n-4))/
       ((n-7)*(n+9)*(n+8)^2*(n+5)^2)))
    end:
seq(a(n), n=7..30);  # Alois P. Heinz, Sep 27 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_] := n! - g[n, 6, {}];
Table[a[n], {n, 7, 25}] (* Jean-François Alcover, Jun 19 2018, from first Maple program *)

a(n) = A214152(n,7) = A000142(n)-A052399(n) = A000142(n)-A214015(n,6). - Alois P. Heinz, Jul 05 2012

Extended beyond a(16) by Alois P. Heinz, Jul 05 2012

A217675 Number of permutations in S_n containing an increasing subsequence of length 8.

Original entry on oeis.org

1, 65, 2603, 83923, 2410291, 64864819, 1683724308, 42918747000, 1086997811325, 27571922195325, 704311698875426, 18189847735254134, 476311846323448840, 12672229166094171240, 343069245941729668380, 9461927811882316662636, 266091066751920438364275

Offset: 8

Alois P. Heinz, Oct 10 2012

nonn

			a(8) = 1: 12345678.

Alois P. Heinz, Table of n, a(n) for n = 8..200

Cf. A000142, A072131, A214015, A214152.

b:= proc(n) option remember; `if`(n<8, n!, ((-343035+429858*n
       +238440*n^3+38958*n^4+634756*n^2+2940*n^5+84*n^6)*b(n-1)
       -(1974*n^4+36336*n^3+213240*n^2+407840*n+82425)*(n-1)^2*b(n-2)
       +2*(49875+42646*n+6458*n^2)*(n-1)^2*(n-2)^2*b(n-3)
       -11025*(n-1)^2*(n-2)^2*(n-3)^2*b(n-4))/ ((n+6)^2*(n+10)^2*(n+12)^2))
    end:
a:= n-> n! -b(n):
seq(a(n), n=8..25);

a(n) = A214152(n,8) = A000142(n)-A072131(n) = A000142(n)-A214015(n,7).

A217676 Number of permutations in S_n containing an increasing subsequence of length 9.

Original entry on oeis.org

1, 82, 4062, 159404, 5497718, 175652924, 5360393100, 159281625000, 4667810722500, 136102249609224, 3973117419487320, 116645785269445696, 3455520662446396976, 103544836992023092832, 3144187412886704149472, 96883566754646092037696, 3032518386648514382974097

Offset: 9

Alois P. Heinz, Oct 10 2012

nonn

			a(9) = 1: 123456789.

Alois P. Heinz, Table of n, a(n) for n = 9..200

Cf. A000142, A072132, A214015, A214152.

b:= proc(n) option remember; `if`(n<4, n!,
      (-147456*(n+4)*(n-1)^2*(n-2)^2*(n-3)^2*b(n-4)
      +128*(33876+30709*n+6687*n^2+410*n^3)*(n-1)^2*(n-2)^2*b(n-3)
      -4*(1092*n^5+37140*n^4+455667*n^3+2387171*n^2+4649270*n+1206000)*
      (n-1)^2*b(n-2) +(-17075520+(22488312+(29223280+(10509820+(1764252+
      (154164+(6804+120*n)*n)*n)*n)*n)*n)*n)*b(n-1))/
      ((n+16)*(n+7)^2*(n+15)^2*(n+12)^2))
    end:
a:= n-> n! -b(n):
seq(a(n), n=9..30);

a(n) = A214152(n,9) = A000142(n)-A072132(n) = A000142(n)-A214015(n,8).

cp's OEIS Frontend

A217057 Number of permutations in S_n containing exactly one increasing subsequence of length 4.

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A269042 Number of permutations of [2n] avoiding the pattern 12...n.

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A072131 T_7(n) in the notation of Bergeron et al., u_k(n) in the notation of Gessel: Related to Young tableaux of bounded height.

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A072132 T_8(n) in the notation of Bergeron et al., u_k(n) in the notation of Gessel: Related to Young tableaux of bounded height.

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A072133 T_9(n) in the notation of Bergeron et al., u_k(n) in the notation of Gessel: Related to Young tableaux of bounded height.

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A158432 Number of permutations of 1..n containing the relative rank sequence { 45312 } at any spacing.

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A159139 Number of permutations of 1..n containing the relative rank sequence { 213465 } at any spacing.

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A159175 Number of permutations of 1..n containing the relative rank sequence { 1234567 } at any spacing.

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