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User: Robert Brignall

Robert Brignall has authored 2 sequences.

A326348 Number of permutations of length n in the class of juxtapositions of separable permutations with 21-avoiders.

1, 1, 2, 6, 24, 115, 609, 3409, 19728, 116692, 701062, 4261581, 26146111, 161631115, 1005522262, 6289410686, 39525228204, 249427451071, 1579885391573, 10040587733693, 64004713573508, 409139527503760, 2622049900367018, 16843666877986873, 108438876033442579

Offset: 0

Robert Brignall, Sep 11 2019

nonn

			There are a(5) = 115 permutations of length 5 which can be expressed as a juxtaposition of a separable permutation (avoiding 2413 and 3142) with an increasing permutation. These 5 cannot be expressed: 25143, 35142, 35241, 41532 and 42531.

Robert Brignall, Jakub Sliacan, Combinatorial specifications for juxtapositions of permutation classes, arXiv:1902.02705 [math.CO], 2019.

Other juxtapositions of algebraic classes with monotone ones are enumerated by A033321, A165538, and A278301.

CoefficientList[Series[(Sqrt[1 - 6*x + x^2]*(2 - 4*x + x^2)*Sqrt[1 - 8*x + 8*x^2]) / (4*(1 - x)*(-2 + 7*x - 7*x^2 + x^3)) + (-10 + 54*x - 99*x^2 + 66*x^3 - 9*x^4 + Sqrt[1 - 6*x + x^2]*(-2 + 10*x - 15*x^2 + 7*x^3) + Sqrt[1 - 8*x + 8*x^2]*(2 - 6*x + x^2 + 6*x^3 - x^4))/(4*(1 - x)^2*(-2 + 7*x - 7*x^2 + x^3)), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 07 2024 *)

G.f.: (2-4*z+z^2)*x*y/(4*(1-z)*(-2+7*z-7*z^2+z^3)) + ((-2+10*z-15*z^2+7*z^3)*x + (2-6*z+z^2+6*z^3-z^4)*y - 10+54*z-99*z^2+66*z^3-9z^4)/(4*(1-z)^2*(-2+7*z-7*z^2+z^3)) where x=sqrt(1-6*z+z^2) and y=sqrt(1-8*z+8z^2).

a(n) ~ (63 + 8*sqrt(2) + 3*sqrt(41 + 40*sqrt(2))) * 2^(3*n/2 - 1) * (1 + sqrt(2))^(n - 1/2) / (sqrt(Pi) * (73 + 53*sqrt(2)) * n^(3/2)). - Vaclav Kotesovec, Jul 07 2024

A326355 Number of permutations of length n with at most two descents.

Original entry on oeis.org

1, 1, 2, 6, 23, 93, 360, 1312, 4541, 15111, 48854, 154674, 482355, 1487905, 4553684, 13857492, 41998265, 126912075, 382702050, 1152300166, 3465813071, 10416313221, 31288785152, 93950241096, 282026883573, 846449748943, 2540120998190, 7621973606682

Offset: 0

Robert Brignall, Sep 11 2019

			For n=4, a(4) = 23 because the permutation 4321 is the only one of length 4 to have more than 2 descents.

D. I. Bevan, On the growth of permutation classes, PhD thesis, The Open University, 2015.
Robert Brignall, Jakub Sliacan, Combinatorial specifications for juxtapositions of permutation classes, arXiv:1902.02705 [math.CO], 2019.
Index entries for linear recurrences with constant coefficients, signature (10,-40,82,-91,52,-12).

Permutations with at most one descent are given by A000325.

Cf. A008292, A123125.

b:= proc(u, o, k) option remember;
      `if`(u+o=0, 1, add(b(u-j, o+j-1, k), j=1..u)+
      `if`(k<2, add(b(u+j-1, o-j, k+1), j=1..o), 0))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..28);  # Alois P. Heinz, Sep 11 2019

LinearRecurrence[{10, -40, 82, -91, 52, -12}, {1, 1, 2, 6, 23, 93}, 30] (* Jean-François Alcover, Mar 01 2020 *)

G.f: 1/(1-z) + z^2/((1-z)^2*(1-2*z)) + z^3*(1+z-4*z^2)/((1-z)^3*(1-2*z)^2*(1-3*z)).
a(n) = Sum_{k=0..3} A123125(n,k). - Alois P. Heinz, Sep 11 2019
a(n) = 3^n -n*2^n +n^2/2 -n/2. - R. J. Mathar, Sep 25 2019

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User: Robert Brignall

A326348 Number of permutations of length n in the class of juxtapositions of separable permutations with 21-avoiders.

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