cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A363809 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, and 2-1-3-5-4.

Original entry on oeis.org

1, 1, 2, 6, 22, 89, 378, 1647, 7286, 32574, 146866, 667088, 3050619, 14039075, 64992280, 302546718, 1415691181, 6656285609, 31436228056, 149079962872, 709680131574, 3390269807364, 16248661836019, 78109838535141, 376531187219762, 1819760165454501
Offset: 0

Views

Author

Andrei Asinowski, Jun 23 2023

Keywords

Comments

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-3-5-4.
The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric pattern "7". See the Merino and Mütze reference, Table 3, entry "12347".

References

  • Andrei Asinowski and Cyril Banderier. Geometry meets generating functions: Rectangulations and permutations (2023).

Links

Crossrefs

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A078482, A033321, A363810, A363811, A363812, A363813, A006012.

Formula

The generating function F=F(x) satisfies the equation x^4*(x - 2)^2*F^4 + x*(x - 2)*(4*x^3 - 7*x^2 + 6*x - 1)*F^3 + (2*x^4 - x^3 - 2*x^2 + 5*x - 1)*F^2 - (4*x^3 - 7*x^2 + 6*x - 1)*F + x^2 = 0.

A363810 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-14-3, and 4-5-3-1-2.

Original entry on oeis.org

1, 1, 2, 6, 21, 79, 306, 1196, 4681, 18308, 71564, 279820, 1095533, 4298463, 16913428, 66769536, 264526329, 1051845461, 4197832133, 16813161765, 67571221016, 272448598737, 1101876945673, 4469106749281, 18174503562880, 74093063050412, 302753929958872
Offset: 0

Views

Author

Andrei Asinowski, Jun 23 2023

Keywords

Comments

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-14-3 and 2-1-3-5-4.
The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "5" and "8". See the Merino and Mütze reference, Table 3, entry "123458".

Links

Crossrefs

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363811, A363812, A363813, A006012.

Programs

  • Maple
    with(gfun): seq(coeff(algeqtoseries(x^8*(-2+x)^2*F^4 - x^3*(x-1)*(-2+x)*(x^5-7*x^4+4*x^3-6*x^2+5*x-1)*F^3 - x*(x-1)*(4*x^7-22*x^6+37*x^5-42*x^4+53*x^3-35*x^2+10*x-1)*F^2 - (5*x^6-16*x^5+15*x^4-28*x^3+23*x^2-8*x+1)*(x-1)^2*F - (2*x^5-5*x^4+4*x^3-10*x^2+6*x-1)*(x-1)^2, x, F, 32, true)[1], x, n+1), n = 0..30); # Vaclav Kotesovec, Jun 24 2023

Formula

The generating function F=F(x) satisfies the equation x^8*(x - 2)^2*F^4 - x^3*(x - 1)*(x - 2)*(x^5 - 7*x^4 + 4*x^3 - 6*x^2 + 5*x - 1)*F^3 - x*(x - 1)*(4*x^7 - 22*x^6 + 37*x^5 - 42*x^4 + 53*x^3 - 35*x^2 + 10*x - 1)*F^2 - (5*x^6 - 16*x^5 + 15*x^4 - 28*x^3 + 23*x^2 - 8*x + 1)*(x - 1)^2*F - (2*x^5 - 5*x^4 + 4*x^3 - 10*x^2 + 6*x - 1)*(x - 1)^2 = 0.

A363811 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-3-5-4, and 4-5-3-1-2.

Original entry on oeis.org

1, 1, 2, 6, 22, 88, 362, 1488, 6034, 24024, 93830, 359824, 1357088, 5043260, 18501562, 67120024, 241169322, 859450004, 3041415520, 10699090888, 37448249502, 130518538696, 453276141238, 1569476495000, 5420784841936, 18683861676756, 64286814548706
Offset: 0

Views

Author

Andrei Asinowski, Jun 23 2023

Keywords

Comments

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-3-5-4 and 4-5-3-1-2.
The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "7" and "8". See the Merino and Mütze reference, Table 3, entry "123478".

Links

Crossrefs

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363812, A363813, A006012.

Programs

  • Mathematica
    CoefficientList[Series[(1 - x)*(1 - 16*x + 109*x^2 - 410*x^3 + 923*x^4 - 1256*x^5 + 988*x^6 - 400*x^7 + 66*x^8 - 2*x^9)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2*(1 - 2*x)^4),{x,0,26}],x] (* Stefano Spezia, Jun 24 2023 *)

Formula

G.f.: (1 - x)*(1 - 16*x + 109*x^2 - 410*x^3 + 923*x^4 - 1256*x^5 + 988*x^6 - 400*x^7 + 66*x^8 - 2*x^9)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2*(1 - 2*x)^4).

A363812 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-4-3, and 3-41-2.

Original entry on oeis.org

1, 1, 2, 6, 20, 69, 243, 870, 3159, 11611, 43130, 161691, 611065, 2325739, 8907360, 34304298, 132770564, 516164832, 2014739748, 7892775473, 31022627947, 122304167437, 483513636064, 1916394053725, 7613498804405, 30313164090695
Offset: 0

Views

Author

Andrei Asinowski, Jun 23 2023

Keywords

Comments

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-4-3 and 3-41-2.
The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "5", "6", "7". See the Merino and Mütze reference, Table 3, entry "1234567".

Links

Crossrefs

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363811, A363813, A006012.

Programs

  • Mathematica
    CoefficientList[Series[(1 - 3*x + 3*x^2 - Sqrt[1 - 6*x + 7*x^2 + 2*x^3 + x^4])/(2*x^2*(2 - x)),{x,0,25}],x] (* Stefano Spezia, Jun 24 2023 *)

Formula

G.f.: (1 - 3*x + 3*x^2 - sqrt(1 - 6*x + 7*x^2 + 2*x^3 + x^4))/(2*x^2*(2 - x)).
Showing 1-4 of 4 results.

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