A116485 -id:A116485 - cp's OEIS frontend

A158423 Number of permutations of 1..n containing the relative rank sequence { 23145 } at any spacing.

1, 26, 459, 7034, 101953, 1454402, 20863742, 304906732, 4578822750, 71092815624, 1146499731400, 19270199314388, 338394827020511, 6218127006568582, 119650035623211443, 2410982527835022898, 50846926185692443237, 1121251553648267523078, 25820703260713964268656, 620088145746453017943268

Offset: 5

R. H. Hardin, Mar 18 2009

nonn

Same series for 43521 12534 54132 35421 31245 12453 54213 43512 23154 21534 45132 35412 31254 21453 45213.

Martin Fuller, Table of n, a(n) for n = 5..38
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A056986, A158005, A116485.

Conjecture: a(n) + A116485(n) = n!. - Benedict W. J. Irwin, Mar 15 2016

Proof: see A116485.

a(17) onwards from A116485, by Martin Fuller, Aug 26 2023

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

A099952 Number of Wilf classes in S_n.

Original entry on oeis.org

1, 1, 1, 3, 16, 91, 595

Offset: 1

N. J. A. Sloane, Nov 12 2004

Z. Stankova and J. West, A new class of Wilf-equivalent permutations, J. Algeb. Combin., 15 (2002), 271-290.

A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
A. M. Baxter and A. D. Jaggard, Pattern avoidance by even permutations, arXiv preprint arXiv:1106.3653, 2011
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152. See Fig. 9.

Representatives for the three Wilf classes in S_4 are A005802, A022558, A061552. - N. J. A. Sloane, Mar 15 2015

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

A256195 Number of permutations in S_n that avoid the pattern 25314.

Original entry on oeis.org

1, 1, 2, 6, 24, 119, 694, 4578, 33184, 258757, 2136978, 18478134, 165857600, 1535336290, 14584260700, 141603589300, 1400942032152, 14087464765300, 143689133196008, 1484090443264936, 15499968503875136, 163501005435759505, 1740170514634463426, 18671118911254798454

Offset: 0

N. J. A. Sloane, Mar 19 2015

nonn

Anthony Guttmann, Table of n, a(n) for n = 0..26
Nathan Clisby, Andrew R. Conway, Anthony J. Guttmann, Yuma Inoue, Classical length-5 pattern-avoiding permutations, arXiv:2109.13485 [math.CO], 2021.
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001.

Representatives for the 16 Wilf-equivalence patterns of length 5 are given in A116485, A047889, and A256195-A256208.

Cf. A099952.

avoid[n_, pat_] := Module[{p1 = pat[[1]], p2 = pat[[2]], p3 = pat[[3]], p4 = pat[[4]], p5 = pat[[5]], lseq = {}, i, p,
    lpat = Subsets[(n + 1) - Range[n], {Length[pat]}],
    psn = Permutations[Range[n]]},
   For[i = 1, i <= Length[lpat], i++,
    p = lpat[[i]];
    AppendTo[lseq, Select[psn, MemberQ[#, {_, p[[p1]], _, p[[p2]], _, p[[p3]], _, p[[p4]], _, p[[p5]], _}, {0}] &]];
    ]; n! - Length[Union[Flatten[lseq, 1]]]];
Table[avoid[n, {2, 5, 3, 1, 4}], {n, 0, 8}]  (* Robert Price, Mar 27 2020 *)

a(14)-a(16) from Bert Dobbelaere, Mar 18 2021

More terms from Anthony Guttmann, Sep 29 2021

A256208 Number of permutations in S_n that avoid the pattern 52341.

Original entry on oeis.org

1, 1, 2, 6, 24, 119, 694, 4582, 33325, 261863, 2192390, 19358590, 178904675, 1720317763, 17132629082, 176055309619, 1861037944163, 20185165186517, 224150069984572, 2543698932578158, 29451619807433107, 347417296695040510, 4170088041714300134, 50874753262007210667

Offset: 0

N. J. A. Sloane, Mar 19 2015

nonn

Anthony Guttmann, Table of n, a(n) for n = 0..23
Nathan Clisby, Andrew R. Conway, Anthony J. Guttmann, Yuma Inoue, Classical length-5 pattern-avoiding permutations, arXiv:2109.13485 [math.CO], 2021.
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001.

Representatives for the 16 Wilf-equivalence patterns of length 5 are given in A116485, A047889, and A256195-A256208.

Cf. A099952.

avoid[n_, pat_] := Module[{p1 = pat[[1]], p2 = pat[[2]], p3 = pat[[3]], p4 = pat[[4]], p5 = pat[[5]], lseq = {}, i, p,
    lpat = Subsets[(n + 1) - Range[n], {Length[pat]}],
    psn = Permutations[Range[n]]},
   For[i = 1, i <= Length[lpat], i++,
    p = lpat[[i]];
    AppendTo[lseq, Select[psn, MemberQ[#, {_, p[[p1]], _, p[[p2]], _, p[[p3]], _, p[[p4]], _, p[[p5]], _}, {0}] &]];
    ]; n! - Length[Union[Flatten[lseq, 1]]]];
Table[avoid[n, {5, 2, 3, 4, 1}], {n, 0, 8}]  (* Robert Price, Mar 27 2020 *)

More terms from Anthony Guttmann, Sep 29 2021

A354511 Number of SAWs crossing a square domain of the hexagonal lattice.

Original entry on oeis.org

2, 14, 264, 21512, 5663596, 6478476233, 23432328776346, 365121393771314359, 18039965927005597824652, 3847346539490622663060402802, 2604549807872636495439504536518768, 7613280873970130888072912524910312775000, 70659728324509466176595292882340210105184200002

Offset: 1

Vaclav Kotesovec, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..24
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D10.

Cf. A001006, A002026, A007764, A116485.

A356610 Number of SAWs crossing a rhomboidal domain of the hexagonal lattice.

Original entry on oeis.org

2, 14, 316, 25092, 7374480, 8029311942, 32223151155864, 476605408516689238, 26016526700583361056456, 5246595079903462547245876694, 3911053741699230141571030313824664, 10780907768757190963361134040036893772360, 109919900687141309301630828947780890728732496678

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D3.

Cf. A001006, A002026, A007764, A116485.

A356616 Number of SAPs crossing a triangular domain of the hexagonal lattice and including top vertex.

Original entry on oeis.org

1, 1, 4, 36, 666, 24696, 1808820, 259300148, 72369408510, 39205936157880, 41152969216872016, 83592236529606631688, 328284931491454739745904, 2490876950205850778116435156, 36494758452603010620499864088198, 1032033208911845667821292289616451218

Offset: 1

Vaclav Kotesovec, following a suggestion from Anthony Guttmann, Aug 16 2022

nonn

Anthony J. Guttmann, Table of n, a(n) for n = 1..26
Anthony J. Guttmann and Iwan Jensen, Self-avoiding walks and polygons crossing a domain on the square and hexagonal lattices, arXiv:2208.06744 [math-ph], Aug 13 2022, Table D9.

Cf. A001006, A002026, A007764, A116485.

A256200 Number of permutations in S_n that avoid the pattern 42351.

Original entry on oeis.org

1, 1, 2, 6, 24, 119, 694, 4580, 33252, 260204, 2161930, 18861307, 171341565, 1610345257, 15579644765, 154541844196, 1566713947713, 16190122718865, 170171678529883, 1816001425551270, 19646035298044543, 215179180467834605, 2383465957654163227, 26673704385975326866

Offset: 0

N. J. A. Sloane, Mar 19 2015

nonn

Anthony Guttmann, Table of n, a(n) for n = 0..27
Nathan Clisby, Andrew R. Conway, Anthony J. Guttmann, Yuma Inoue, Classical length-5 pattern-avoiding permutations, arXiv:2109.13485 [math.CO], 2021.
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001.

Representatives for the 16 Wilf-equivalence patterns of length 5 are given in A116485, A047889, and A256195-A256208.

Cf. A099952, A158434.

avoid[n_, pat_] := Module[{p1 = pat[[1]], p2 = pat[[2]], p3 = pat[[3]], p4 = pat[[4]], p5 = pat[[5]], lseq = {}, i, p,
    lpat = Subsets[(n + 1) - Range[n], {Length[pat]}],
    psn = Permutations[Range[n]]},
   For[i = 1, i <= Length[lpat], i++,
    p = lpat[[i]];
    AppendTo[lseq, Select[psn, MemberQ[#, {_, p[[p1]], _, p[[p2]], _, p[[p3]], _, p[[p4]], _, p[[p5]], _}, {0}] &]];
    ]; n! - Length[Union[Flatten[lseq, 1]]]];
Table[avoid[n, {4, 2, 3, 5, 1}], {n, 0, 8}]  (* Robert Price, Mar 27 2020 *)

a(n) = n! - A158434(n). - Andrew Howroyd, May 18 2020

a(14)-a(15) added by Andrew Howroyd, May 18 2020

More terms from Anthony Guttmann, Sep 29 2021

A256196 Number of permutations in S_n that avoid the pattern 31524.

Original entry on oeis.org

1, 1, 2, 6, 24, 119, 694, 4579, 33216, 259401, 2147525, 18632512, 167969934, 1563027614, 14937175825, 146016423713, 1455402205257, 14753501614541, 151783381341695, 1582029822426003, 16681492660789425, 177726496203056670, 1911230701872865231, 20726637978574528119

Offset: 0

N. J. A. Sloane, Mar 19 2015

nonn

Anthony Guttmann, Table of n, a(n) for n = 0..25
Nathan Clisby, Andrew R. Conway, Anthony J. Guttmann, Yuma Inoue, Classical length-5 pattern-avoiding permutations, arXiv:2109.13485 [math.CO], 2021.
Zvezdelina Stankova-Frenkel and Julian West, A new class of Wilf-equivalent permutations, arXiv:math/0103152 [math.CO], 2001.

Representatives for the 16 Wilf-equivalence patterns of length 5 are given in A116485, A047889, and A256195-A256208.

Cf. A099952.

avoid[n_, pat_] := Module[{p1 = pat[[1]], p2 = pat[[2]], p3 = pat[[3]], p4 = pat[[4]], p5 = pat[[5]], lseq = {}, i, p,
    lpat = Subsets[(n + 1) - Range[n], {Length[pat]}],
    psn = Permutations[Range[n]]},
   For[i = 1, i <= Length[lpat], i++,
    p = lpat[[i]];
    AppendTo[lseq, Select[psn, MemberQ[#, {_, p[[p1]], _, p[[p2]], _, p[[p3]], _, p[[p4]], _, p[[p5]], _}, {0}] &]];
    ]; n! - Length[Union[Flatten[lseq, 1]]]];
Table[avoid[n, {3, 1, 5, 2, 4}], {n, 0, 8}]  (* Robert Price, Mar 27 2020 *)

a(14)-a(16) from Bert Dobbelaere, Mar 18 2021

More terms from Anthony Guttmann, Sep 29 2021

cp's OEIS Frontend

A158423 Number of permutations of 1..n containing the relative rank sequence { 23145 } at any spacing.

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

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A099952 Number of Wilf classes in S_n.

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A256195 Number of permutations in S_n that avoid the pattern 25314.

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A256208 Number of permutations in S_n that avoid the pattern 52341.

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A354511 Number of SAWs crossing a square domain of the hexagonal lattice.

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A356610 Number of SAWs crossing a rhomboidal domain of the hexagonal lattice.

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A356616 Number of SAPs crossing a triangular domain of the hexagonal lattice and including top vertex.

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A256200 Number of permutations in S_n that avoid the pattern 42351.

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Author

Keywords

Links

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Programs

Mathematica

Formula

Extensions

A256196 Number of permutations in S_n that avoid the pattern 31524.

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