A191830 -id:A191830 - cp's OEIS frontend

A226432 The number of simple permutations of length n in a particular geometric grid class.

1, 2, 0, 2, 3, 7, 13, 25, 46, 84, 151, 269, 475, 833, 1452, 2518, 4347, 7475, 12809, 21881, 37274, 63336, 107375, 181657, 306743, 517057, 870168, 1462250, 2453811, 4112479, 6884101, 11510809, 19226950, 32084028, 53489287, 89097893, 148290067, 246615425, 409835844, 680609086

Offset: 1

Jay Pantone, Jun 06 2013

This geometric grid class is given by the array [[0,0,1,0],[0,0,0,1],[0,1,-1,0],[1,0,0,-1]]. A picture is given in the LINKS section.

The sequence of all permutations in this class is given by A226431.

Jay Pantone, The Enumeration of Permutations Avoiding 3124 and 4312, arXiv:1309.0832 [math.CO], (2013).
Jay Pantone, Picture of the geometric grid class
Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Join[{1, 2}, LinearRecurrence[{2, 1, -2, -1}, {0, 2, 3, 7}, 40]] (* Jean-François Alcover, Jul 21 2018 *)

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

G.f.: x+2*x^2+ x^4*(1-x)*(2+x)/(1-x-x^2)^2 (corrected, Joerg Arndt, Jun 26 2013)

a(n) = A191830(n+2)-A000045(n+2), n>=4. - R. J. Mathar, Aug 31 2013

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

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

Original entry on oeis.org

1, 1, 2, 2, 2, 2, 2, 3, 3, 4, 4, 4, 4, 3, 3, 4, 4, 6, 7, 8, 8, 8, 7, 6, 4, 4, 5, 5, 8, 10, 13, 15, 16, 16, 15, 13, 10, 8, 5, 5, 6, 6, 10, 13, 18, 23, 28, 31, 32, 31, 28, 23, 18, 13, 10, 6, 6, 7, 7, 12, 16, 23, 31, 41, 51, 59, 63, 63, 59, 51, 41, 31, 23, 16, 12, 7, 7

Offset: 1

Lechoslaw Ratajczak, Jul 04 2020

The number of terms in row n is 3*n-1 = A016789(n-1).
The sum of row n is equal to 2*A094002(n-1) = 2*A188589(n).
Fibonacci(n) = T(n+k,n) - T(n+k-1,n) for n >= 1, k = 1,2,3,...
The elements b(k) of the main diagonal, superdiagonal 1 and all subdiagonals have the recursive formula: b(k) = 2*b(k-1) + b(k-2) - 2*b(k-3) - b(k-4) for k > 4.

			Triangle begins:
n\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20...
1   1  1
2   2  2  2  2  2
3   3  3  4  4  4  4  3  3
4   4  4  6  7  8  8  8  7  6  4  4
5   5  5  8 10 13 15 16 16 15 13 10  8  5  5
6   6  6 10 13 18 23 28 31 32 31 28 23 18 13 10  6  6
7   7  7 12 16 23 31 41 51 59 63 63 59 51 41 31 23 16 12  7  7
...

Superdiagonal 1 is A029907 for n >= 1.
The main diagonal is A208354 for n >= 1.
Subdiagonal 1 is A102702(n-1) for n >= 1.
Subdiagonal 2 is A206268(n+2) for n >= 1 (conjectured).
Subdiagonal 3 is A191830(n+3) for n >= 1.
Cf. A007318, A051597, A228196.

T(n,k) = T(n,3*k-n) for 1 <= k <= 3*n-1.
T(n,k) = Sum_{u=2*(n-k)+3..2*n-k+1} ceiling(u/2)*A065941(k-2,u-2*(n-k)-3) for n >= 3, 3 <= k <= n.
T(n,k) = Sum_{m1=1..k-n} A208354(m1)*binomial(n-m1-1, k-n-m1) + Sum_{m2=1..2*n-k} A208354(m2)*binomial(n-m2-1, 2*n-k-m2) for n >= 2, n+1 <= k <= 2*n-1.
T(n,k) = Sum_{u=2*(k-2*n)+3..k-n+1} ceiling(u/2)*A065941(3*n-k-2,u-2*(k-2*n)-3) for n>= 3, 2*n <= k <= 3*(n-1).
T(n,k) = A208354(k) + (n-k)*Fibonacci(k) for n >= 3, 3 <= k <= n.
T(n,k) = A029907(k-1) + (n-k+1)*Fibonacci(k) for n >= 2, 3 <= k <= n+1.

cp's OEIS Frontend

A226432 The number of simple permutations of length n in a particular geometric grid class.

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

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Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

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cp's OEIS Frontend

A226432 The number of simple permutations of length n in a particular geometric grid class.

Views

Author

Keywords

Comments

Links

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3*n-2) = T(n,3*n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).

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Author

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Comments

Examples

Crossrefs

Formula

A336014 Irregular triangle read by rows: T(n,1) = T(n,2) = T(n,3n-2) = T(n,3n-1) = n for n >= 1 and T(n,k) = T(n-1,k-2) + T(n-1,k-1) for n > 1, 3 <= k <= 3*(n-1).