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 1-8 of 8 results.

A361272 Number of 1243-avoiding even Grassmannian permutations of size n.

Original entry on oeis.org

1, 1, 1, 3, 6, 12, 20, 32, 47, 67, 91, 121, 156, 198, 246, 302, 365, 437, 517, 607, 706, 816, 936, 1068, 1211, 1367, 1535, 1717, 1912, 2122, 2346, 2586, 2841, 3113, 3401, 3707, 4030, 4372, 4732, 5112, 5511, 5931, 6371, 6833, 7316, 7822, 8350, 8902, 9477, 10077
Offset: 0

Views

Author

Juan B. Gil, Mar 09 2023

Keywords

Comments

A permutation is said to be Grassmannian if it has at most one descent. A permutation is even if it has an even number of inversions.
a(n) is also the number of sigma-avoiding even Grassmannian permutations of size n, where sigma is any of the patterns 2134, 2341, or 4123.

Examples

			For n=4 the a(4) = 6 permutations are 1234, 1342, 1423, 2314, 3124, 3412.

Links

Crossrefs

Cf. A175287, A356185, A361273.

Formula

G.f.: -(2*x^4-4*x^3+2*x-1)/((x+1)*(x-1)^4).
a(n) = (57 - 9*(-1)^n - 28*n + 6*n^2 + 4*n^3)/48. - Stefano Spezia, Mar 09 2023

A361270 Number of 1324-avoiding odd Grassmannian permutations of size n.

Original entry on oeis.org

0, 0, 1, 2, 5, 8, 16, 20, 38, 40, 75, 70, 131, 112, 210, 168, 316, 240, 453, 330, 625, 440, 836, 572, 1090, 728, 1391, 910, 1743, 1120, 2150, 1360, 2616, 1632, 3145, 1938, 3741, 2280, 4408, 2660, 5150, 3080, 5971, 3542, 6875, 4048, 7866, 4600, 8948, 5200, 10125
Offset: 0

Views

Author

Juan B. Gil, Mar 07 2023

Keywords

Comments

A permutation is said to be Grassmannian if it has at most one descent. A permutation is odd if it has an odd number of inversions.

Examples

			For n=4 the a(4)=5 permutations are 1243, 2134, 2341, 2413, 4123.

Links

Crossrefs

Cf. A356185, A361271.

Programs

  • PARI
    Vec(x^2*(2*x^4+x^2+2*x+1)/((1+x)^4*(1-x)^4)+O(x^50)) \\ Michel Marcus, Mar 07 2023

Formula

G.f.: x^2*(2*x^4+x^2+2*x+1)/((1+x)^4*(1-x)^4).

A361271 Number of 1342-avoiding odd Grassmannian permutations of size n.

Original entry on oeis.org

0, 0, 1, 2, 6, 9, 19, 25, 44, 54, 85, 100, 146, 167, 231, 259, 344, 380, 489, 534, 670, 725, 891, 957, 1156, 1234, 1469, 1560, 1834, 1939, 2255, 2375, 2736, 2872, 3281, 3434, 3894, 4065, 4579, 4769, 5340, 5550, 6181, 6412, 7106, 7359, 8119, 8395, 9224, 9524, 10425
Offset: 0

Views

Author

Juan B. Gil, Mar 07 2023

Keywords

Comments

A permutation is said to be Grassmannian if it has at most one descent. A permutation is odd if it has an odd number of inversions.
a(n) is also the number of 3124-avoiding odd Grassmannian permutations of size n.

Examples

			For n=4 the a(4)=6 permutations are 1243, 1324, 2134, 2341, 2413, 4123.

Links

Crossrefs

Cf. A356185, A361270, A361274.

Programs

  • PARI
    seq(n) = Vec(x^2*(x^4+x^2+x+1)/((1+x)^3*(1-x)^4) + O(x*x^n), -n-1) \\ Andrew Howroyd, Mar 07 2023

Formula

G.f.: x^2*(x^4+x^2+x+1)/((1+x)^3*(1-x)^4).

A361273 Number of 1324-avoiding even Grassmannian permutations of size n.

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 20, 37, 47, 81, 91, 151, 156, 253, 246, 393, 365, 577, 517, 811, 706, 1101, 936, 1453, 1211, 1873, 1535, 2367, 1912, 2941, 2346, 3601, 2841, 4353, 3401, 5203, 4030, 6157, 4732, 7221, 5511, 8401, 6371, 9703, 7316, 11133, 8350, 12697, 9477, 14401, 10701
Offset: 0

Views

Author

Juan B. Gil, Mar 09 2023

Keywords

Comments

A permutation is said to be Grassmannian if it has at most one descent. A permutation is even if it has an even number of inversions.

Examples

			For n=4 the a(4) = 6 permutations are 1234, 1342, 1423, 2314, 3124, 3412.

Links

Crossrefs

Cf. A356185, A361270, A361272.

Formula

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

A361274 Number of 1342-avoiding even Grassmannian permutations of size n.

Original entry on oeis.org

1, 1, 1, 3, 5, 12, 17, 32, 41, 67, 81, 121, 141, 198, 225, 302, 337, 437, 481, 607, 661, 816, 881, 1068, 1145, 1367, 1457, 1717, 1821, 2122, 2241, 2586, 2721, 3113, 3265, 3707, 3877, 4372, 4561, 5112, 5321, 5931, 6161, 6833, 7085, 7822, 8097, 8902, 9201, 10077, 10401
Offset: 0

Views

Author

Juan B. Gil, Mar 09 2023

Keywords

Comments

A permutation is said to be Grassmannian if it has at most one descent. A permutation is even if it has an even number of inversions.
a(n) is also the number of 3124-avoiding even Grassmannian permutations of size n.

Examples

			For n=4 the a(4) = 5 permutations are 1234, 1423, 2314, 3124, 3412.

Links

Crossrefs

Cf. A356185, A361271.

Programs

  • Mathematica
    LinearRecurrence[{1,3,-3,-3,3,1,-1},{1,1,1,3,5,12,17},51] (* Stefano Spezia, Mar 09 2023 *)

Formula

G.f.: -(2*x^6-x^5-5*x^4-2*x^3+3*x^2-1)/((x+1)^3*(x-1)^4).
E.g.f.: ((24 - 9*x + 6*x^2 + 2*x^3)*cosh(x) + (33 - 6*x + 9*x^2 + 2*x^3)*sinh(x))/24. - Stefano Spezia, Mar 09 2023

A361275 Number of 1423-avoiding even Grassmannian permutations of size n.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 17, 29, 41, 61, 81, 111, 141, 183, 225, 281, 337, 409, 481, 571, 661, 771, 881, 1013, 1145, 1301, 1457, 1639, 1821, 2031, 2241, 2481, 2721, 2993, 3265, 3571, 3877, 4219, 4561, 4941, 5321, 5741, 6161, 6623, 7085, 7591, 8097, 8649, 9201, 9801, 10401
Offset: 0

Views

Author

Juan B. Gil, Mar 10 2023

Keywords

Comments

A permutation is said to be Grassmannian if it has at most one descent. A permutation is even if it has an even number of inversions.
Avoiding any of the patterns 2314 or 3412 gives the same sequence.

Examples

			For n=4 the a(4) = 5 permutations are 1234, 1342, 2314, 3124, 3412.

Links

Crossrefs

Cf. A356185, A361272, A361273, A361274.
For the corresponding odd permutations, cf. A005993.

Programs

  • Maple
    seq(1 - 5*n/24 + n^3/12 - (-1)^n * n/8, n = 0 .. 100); # Robert Israel, Mar 10 2023

Formula

G.f.: -(x^5-x^4-4*x^3+2*x^2+x-1)/((x+1)^2*(x-1)^4).
a(n) = 1 - 5*n/24 + n^3/12 - (-1)^n * n/8. - Robert Israel, Mar 10 2023

A362193 Number of Grassmannian permutations of size n that avoid a pattern, sigma, where sigma is a pattern of size 6 with exactly one descent.

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 57, 113, 211, 373, 628, 1013, 1574, 2367, 3459, 4929, 6869, 9385, 12598, 16645, 21680, 27875, 35421, 44529, 55431, 68381, 83656, 101557, 122410, 146567, 174407, 206337, 242793, 284241, 331178, 384133, 443668, 510379, 584897
Offset: 0

Views

Author

Jessica A. Tomasko, Apr 10 2023

Keywords

Comments

A permutation is said to be Grassmannian if it has at most one descent. The definition for sigma is a pattern of size 6 with exactly one descent. For example, sigma can be chosen to be 124356, 241356, 361245, 512346, etc.

Links

Crossrefs

Cf. A027660, A361270, A361271, A356185, A000325, A027927.

Programs

  • Maple
    a:= n-> 1+(n-1)*n*(n+1)*(n*(n-5)+26)/120:
seq(a(n), n=0..38);  # Alois P. Heinz, Apr 12 2023
  • Mathematica
    CoefficientList[Series[(1 - 5 x + 11 x^2 - 12 x^3 + 7 x^4 - x^5)/(1 - x)^6, {x, 0, 38}], x] (* Michael De Vlieger, Apr 12 2023 *)
  • PARI
    a(n) = 1 + sum(i=3, 6, binomial(n, i-1)) \\ Andrew Howroyd, Apr 10 2023

Formula

a(n) = 1 + Sum_{i=2..5} binomial(n,i).
G.f.: (1-5*x+11*x^2-12*x^3+7*x^4-x^5)/(1-x)^6.
a(0) = 1; a(1) = 1; a(n) = 1 + A027660(n-2), n >= 2. - Omar E. Pol, Apr 12 2023

A361276 Number of 2413-avoiding even Grassmannian permutations of size n.

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 22, 37, 55, 81, 111, 151, 196, 253, 316, 393, 477, 577, 685, 811, 946, 1101, 1266, 1453, 1651, 1873, 2107, 2367, 2640, 2941, 3256, 3601, 3961, 4353, 4761, 5203, 5662, 6157, 6670, 7221, 7791, 8401, 9031, 9703, 10396, 11133, 11892, 12697, 13525, 14401
Offset: 0

Views

Author

Juan B. Gil, Mar 10 2023

Keywords

Comments

A permutation is said to be Grassmannian if it has at most one descent. A permutation is even if it has an even number of inversions.

Examples

			For n=4 the a(4) = 6 permutations are 1234, 1342, 1423, 2314, 3124, 3412.

Links

Crossrefs

Cf. A356185, A361272, A361273, A361274, A361275.
For the corresponding odd permutations, cf. A006918.

Programs

  • Mathematica
    LinearRecurrence[{2,1,-4,1,2,-1},{1,1,1,3,6,13},50] (* Harvey P. Dale, Aug 14 2023 *)

Formula

G.f.: -(x^5-2*x^4-4*x^3+2*x^2+x-1)/((x+1)^2*(x-1)^4).
Showing 1-8 of 8 results.

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