A165534 -id:A165534 - cp's OEIS frontend

A228769 The number of skew sum decomposable permutations which avoid the patterns 3124 and 4312.

0, 1, 3, 10, 35, 129, 494, 1935, 7670, 30582, 122280, 489552, 1960956, 7855994, 31471731, 126063782, 504888839, 2021777865, 8094784697, 32405289263, 129709206465, 519129580361, 2077477804103, 8313000733125, 33261722967167, 133076495664483, 532391828669675, 2129796460981743, 8519701993370619, 34079469569317323

Offset: 1

Jay Pantone, Sep 08 2013

nonn

			Example: a(4)=10 because there are 10 skew sum decomposable permutations of length 4 which avoid the patterns 3124 and 4312.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jay Pantone, The Enumeration of Permutations Avoiding 3124 and 4312, arXiv:1309.0832 [math.CO], (2013)

The class of all permutations which avoid the patterns 3124 and 4312 is given by A165534.

CoefficientList[Series[- (1/x) (3 x^4 - x^3 + Sqrt[-4 x + 1] (4 x^5 - 9 x^4 + 9 x^3 - 2 x^2)) / (12 x^4 - 31 x^3 + 27 x^2 + Sqrt[-4 x + 1] (4 x^4 - 13 x^3 + 15 x^2 - 7 x + 1) - 9 x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 09 2013 *)

G.f.: -(3*x^4 - x^3 + sqrt(-4*x + 1)*(4*x^5 - 9*x^4 + 9*x^3 - 2*x^2))/(12*x^4 - 31*x^3 + 27*x^2 + sqrt(-4*x + 1)*(4*x^4 - 13*x^3 + 15*x^2 - 7*x + 1) - 9*x + 1).

a(n) ~ 4^(n-1)/9 * (1 + 1/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 18 2014

A226430 The number of simple permutations of length n which avoid 1243 and 2431.

Original entry on oeis.org

1, 2, 0, 2, 4, 10, 21, 44, 89, 178, 352, 692, 1355, 2648, 5171, 10100, 19744, 38646, 75761, 148772, 292653, 576678, 1138240, 2250152, 4454679, 8830640, 17525991, 34820264, 69244864, 137815978, 274487517, 547035452, 1090790465, 2176043098, 4342753696, 8669805020, 17313228899

Offset: 1

Jay Pantone, Jun 06 2013

nonn

Jay Pantone, The Enumeration of Permutations Avoiding 3124 and 4312, arXiv:1309.0832 [math.CO], (2013)
Wikipedia, Permutation classes avoiding two patterns of length 4
Index entries for linear recurrences with constant coefficients, signature (4,-3,-4,3,2).

The number of all permutations which avoid 1243 and 2431 is A165534.

Join[{1, 2}, LinearRecurrence[{4, -3, -4, 3, 2}, {0, 2, 4, 10, 21}, 40]] (* Jean-François Alcover, Jul 22 2018 *)

x='x+O('x^66); Vec((x-2*x^2-5*x^3+12*x^4+x^5-8*x^6-3*x^7)/((1-2*x)*(1-x-x^2)^2)) \\ Joerg Arndt, Jun 19 2013

G.f.: (x-2*x^2-5*x^3+12*x^4+x^5-8*x^6-3*x^7)/((1-2*x)*(1-x-x^2)^2).

a(n) = -2*A000045(n+1) +A191830(n+2) +2^(n-3), n>2. - R. J. Mathar, Dec 06 2013

A228770 The number of sum indecomposable permutations which avoid the patterns 3124 and 4312.

Original entry on oeis.org

1, 1, 3, 12, 51, 217, 912, 3785, 15554, 63458, 257566, 1041548, 4200462, 16906262, 67943341, 272740788, 1093881967, 4384217569, 17562176283, 70319782015, 281466691159, 1126304935761, 4505961410365, 18023526090613, 72082118816829, 288245594631227, 1152536796877409, 4607992736095739, 18422141293792669, 73645313049839723

Offset: 1

Jay Pantone, Sep 08 2013

nonn

			Example: a(4)=12 because there are 12 sum indecomposable permutations of length 4 which avoid the patterns 3124 and 4312.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jay Pantone, The Enumeration of Permutations Avoiding 3124 and 4312, arXiv:1309.0832 [math.CO], (2013)

A228770(n) = A165534(n) - A226434(n)

CoefficientList[Series[- (1/x) (24 x^6 - 71 x^5 + 84 x^4 - 45 x^3 + 11 x^2 + Sqrt[-4 x + 1] (4 x^6 - 25 x^5 + 40 x^4 - 29 x^3 + 9 x^2 - x) - x) / (8 x^6 - 54 x^5 + 117 x^4 - 114 x^3 + 54 x^2 - Sqrt[-4 x + 1] (12 x^5 - 43 x^4 + 58 x^3 - 36 x^2 + 10 x - 1) - 12 x + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 09 2013 *)

G.f.: -(24*x^6 - 71*x^5 + 84*x^4 - 45*x^3 + 11*x^2 + sqrt(-4*x + 1)*(4*x^6 - 25*x^5 + 40*x^4 - 29*x^3 + 9*x^2 - x) - x)/(8*x^6 - 54*x^5 + 117*x^4 - 114*x^3 + 54*x^2 - sqrt(-4*x + 1)*(12*x^5 - 43*x^4 + 58*x^3 - 36*x^2 + 10*x - 1) - 12*x + 1).
a(n) ~ 2^(2*n-1)/9 * (1+2/(sqrt(Pi*n))). - Vaclav Kotesovec, Mar 20 2014
Conjecture: -(n+1)*(39961*n-2474598)*a(n) +(-39961*n^2-25975201*n+4949196) *a(n-1) +3*(1460811*n^2+27429105*n-41310802) *a(n-2) +3 *(-8653921*n^2-4750029*n+74360724) *a(n-3) +4*(15005713*n^2-82481258*n+83094771) *a(n-4) +12*(-4937548*n^2+40726604*n-73155719) *a(n-5) +16*(652718*n-2110173)*(2*n-13) *a(n-6)=0. - R. J. Mathar, Jun 14 2016

A228771 The number of skew sum indecomposable permutations which avoid the patterns 3124 and 4312.

Original entry on oeis.org

1, 1, 3, 12, 53, 234, 1013, 4306, 18051, 74903, 308487, 1263393, 5152139, 20941298, 84897207, 343467388, 1387244237, 5595368133, 22543241377, 90739796783, 364954106877, 1466865660103, 5892463315373, 23658818086719, 94952826295865, 380947979933041, 1527871081396065, 6126157580638517, 24557525359295337, 98421154766829972

Offset: 1

Jay Pantone, Sep 08 2013

nonn

			Example: a(4)=12 because there are 12 skew sum indecomposable permutations of length 4 which avoid the patterns 3124 and 4312.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jay Pantone, The Enumeration of Permutations Avoiding 3124 and 4312, arXiv:1309.0832 [math.CO], (2013)

A228771(n) = A165534(n) - A228769(n)

CoefficientList[Series[(1/x) (8 x^6 - 28 x^5 + 50 x^4 - 35 x^3 + 10 x^2 - Sqrt[-4 x + 1] (6 x^5 - 18 x^4 + 21 x^3 - 8 x^2 + x) - x) / (8 x^5 - 46 x^4 + 71 x^3 - 43 x^2 - Sqrt[-4 x + 1] (12 x^4 - 31 x^3 + 27 x^2 - 9 x + 1) + 11 x - 1), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 09 2013 *)

G.f.: (8*x^6 - 28*x^5 + 50*x^4 - 35*x^3 + 10*x^2 - sqrt(-4*x + 1)*(6*x^5 - 18*x^4 + 21*x^3 - 8*x^2 + x) - x)/(8*x^5 - 46*x^4 + 71*x^3 - 43*x^2 - sqrt(-4*x + 1)*(12*x^4 - 31*x^3 + 27*x^2 - 9*x + 1) + 11*x - 1).
a(n) ~ 4^(n-1)/3 * (1+1/sqrt(Pi*n)). - Vaclav Kotesovec, Mar 20 2014
Conjecture: -163*(n+2)*(4*n-413) *a(n) +(-652*n^2-725425*n-452889) *a(n-1) +5*(14473*n^2+512276*n-443094) *a(n-2) +(-410045*n^2-2408964*n+8429009) *a(n-3) +2*(404156*n^2-1297075*n-1518084)*a(n-4) -8*(29333*n-32490)*(2*n-11)*a(n-5)=0. - R. J. Mathar, Jun 14 2016

Corrected a(17) by Vincenzo Librandi, Sep 09 2013

cp's OEIS Frontend

A228769 The number of skew sum decomposable permutations which avoid the patterns 3124 and 4312.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A226430 The number of simple permutations of length n which avoid 1243 and 2431.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A228770 The number of sum indecomposable permutations which avoid the patterns 3124 and 4312.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A228771 The number of skew sum indecomposable permutations which avoid the patterns 3124 and 4312.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions