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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.

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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

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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

A230447 T(n, k) = T(n-1, k) + T(n-1, k-1) + A230135(n, k) with T(n, 0) = A008619(n) and T(n, n) = A080239(n+1), n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 4, 5, 3, 3, 6, 9, 8, 6, 3, 9, 16, 17, 14, 9, 4, 12, 25, 33, 32, 23, 15, 4, 16, 38, 58, 65, 55, 39, 24, 5, 20, 54, 96, 124, 120, 94, 63, 40, 5, 25, 75, 150, 220, 244, 215, 157, 103, 64, 6, 30, 100, 225, 371, 464, 459, 372, 261, 167, 104
Offset: 0

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Author

Johannes W. Meijer, Oct 19 2013

Keywords

Comments

The terms in the right hand columns of triangle T(n, k) and the terms in the rows of the square array Tsq(n, k) represent the Kn1p sums of the ‘Races with Ties’ triangle A035317.
For the definitions of the Kn1p sums see A180662. This sequence is related to A230448.
The first few row sums are: 1, 2, 6, 14, 32, 68, 144, 299, 616, 1258, 2559, 5185, 10478, … .

Examples

			The first few rows of triangle T(n, k) n >= 0 and 0 <= k <= n.
n/k 0   1   2    3    4     5     6     7
------------------------------------------------
0|  1
1|  1,  1
2|  2,  2,  2
3|  2,  4,  5,   3
4|  3,  6,  9,   8,   6
5|  3,  9, 16,  17,  14,    9
6|  4, 12, 25,  33,  32,   23,    15
7|  4, 16, 38,  58,  65,   55,    39,   24
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
n/k 0   1   2    3    4     5     6     7
------------------------------------------------
0|  1,  1,  2,   3,   6,    9,   15,   24
1|  1,  2,  5,   8,  14,   23,   39,   63
2|  2,  4,  9,  17,  32,   55,   94,  157
3|  2,  6, 16,  33,  65,  120,  215,  372
4|  3,  9, 25,  58, 124,  244,  459,  831
5|  3, 12, 38,  96, 220,  464,  924, 1755
6|  4, 16, 54, 150, 371,  835, 1759, 3514
7|  4, 20, 75, 225, 596, 1431, 3191, 6705

Crossrefs

Cf. (Triangle columns) A008619, A002620, A175287, A080239
Cf. A035317, A230448, A230449, A230135, A080239, A034851, A228570

Programs

  • Maple
    T := proc(n, k): add(A035317(n-i, n-k+i), i=0..floor(k/2)) end: A035317 := proc(n, k): add((-1)^(i+k) * binomial(i+n-k+1, i), i=0..k) end: seq(seq(T(n, k), k=0..n), n=0..10); # End first program.
T := proc(n, k) option remember: if k=0 then return(A008619(n)) elif k=n then return(A080239(n+1)) else A230135(n, k) + procname(n-1, k) + procname(n-1, k-1) fi: end: A008619 := n -> floor(n/2) +1: A080239 := n -> add(combinat[fibonacci](n-4*k), k=0..floor((n-1)/4)): A230135 := proc(n, k): if ((k mod 4 = 2) and (n mod 2 = 1)) or ((k mod 4 = 0) and (n mod 2 = 0)) then return(1) else return(0) fi: end: seq(seq(T(n, k), k=0..n), n=0..10); # End second program.

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + A230135(n, k) with T(n, 0) = A008619(n) and T(n, n) = A080239(n+1), n >= 0 and 0 <= k <= n.
T(n, k) = sum(A035317(n-i, n-k+i), i = 0..floor(k/2)), n >= 0 and 0 <= k <= n.
The triangle as a square array Tsq(n, k) = T(n+k, k), n >= 0 and k >= 0.
Tsq(n, k) = sum(A035317(n+k-i, n+i), i=0..floor(k/2)), n >= 0 and k >= 0.
Tsq(n, k) = A080239(2*n+k+1) - sum(A035317(2*n+k-i, i), i=0..n-1).
The G.f. generates the terms in the n-th row of the square array Tsq(n, k).
G.f.: a(n)/(4*(x-1)) + 1/(4*(x+1)) + (-1)^n*(x+2)/(10*(x^2+1)) - (A000032(2*n+3) + A000032(2*n+2)*x)/(5*(x^2+x-1)) + sum((-1)^(k+1) * A064831(n-k+1)/((x-1)^k), k= 2..n), n >= 0, with a(n) = A064831(n+1) + 2*A064831(n) - 2*A064831(n-1) + A064831(n-2).

A282044 Reduced Kronecker coefficients for the case a=2, b=3, i=4.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 9, 16, 24, 37, 51, 71, 93
Offset: 0

Views

Author

N. J. A. Sloane, Feb 21 2017

Keywords

Comments

Table 3 of Colmenarejo (2016) shows this sequence as the missing member of the family A266769, A000601, A006918, A014126, A282044, A175287.
It would be nice to have a g.f.

Links

Crossrefs

Cf. A266769, A000601, A006918, A014126, A175287.

Formula

Conjectured g.f.: x^4*(1+x^2)/(1-2*x+x^3+3*x^5-4*x^6). - Jean-François Alcover, Feb 18 2019.
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