A361270 -id:A361270 - cp's OEIS frontend

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

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

Juan B. Gil, Mar 07 2023

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.

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

Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Cf. A356185, A361270, A361274.

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

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

Juan B. Gil, Mar 09 2023

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.

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

Juan B. Gil and Jessica A. Tomasko, Pattern-avoiding even and odd Grassmannian permutations, arXiv:2207.12617 [math.CO], 2022.
Index entries for linear recurrences with constant coefficients, signature (0,4,0,-6,0,4,0,-1).

Cf. A356185, A361270, A361272.

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).

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

Jessica A. Tomasko, Apr 10 2023

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.

Juan B. Gil and Jessica A. Tomasko, Restricted Grassmannian permutations, Enum. Combin. Appl. 2 (2022), no. 4, Article #S4PP6.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

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

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

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 *)

a(n) = 1 + sum(i=3, 6, binomial(n, i-1)) \\ Andrew Howroyd, Apr 10 2023

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

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

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

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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.

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