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A260665
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Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 12-3; triangle T(n,k), n>=0, 0<=k<=(n-1)*(n-2)/2-[n=0], read by rows.
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14
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1, 1, 2, 5, 1, 15, 7, 1, 1, 52, 39, 13, 12, 2, 1, 1, 203, 211, 112, 103, 41, 24, 17, 5, 2, 1, 1, 877, 1168, 843, 811, 492, 337, 238, 122, 68, 39, 28, 8, 5, 2, 1, 1, 4140, 6728, 6089, 6273, 4851, 3798, 2956, 1960, 1303, 859, 594, 314, 204, 110, 64, 43, 17, 8, 5, 2, 1, 1
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OFFSET
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0,3
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COMMENTS
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Patterns 1-23, 3-21, 32-1 give the same triangle.
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LINKS
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FORMULA
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EXAMPLE
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T(4,1) = 7: 1324, 1342, 2134, 2314, 2341, 3124, 4123.
T(4,2) = 1: 1243.
T(4,3) = 1: 1234.
T(5,3) = 12: 12534, 12543, 13245, 13425, 13452, 21345, 23145, 23415, 23451, 31245, 41235, 51234.
T(5,4) = 2: 12435, 12453.
T(5,5) = 1: 12354.
T(5,6) = 1: 12345.
Triangle T(n,k) begins:
0 : 1;
1 : 1;
2 : 2;
3 : 5, 1;
4 : 15, 7, 1, 1;
5 : 52, 39, 13, 12, 2, 1, 1;
6 : 203, 211, 112, 103, 41, 24, 17, 5, 2, 1, 1;
7 : 877, 1168, 843, 811, 492, 337, 238, 122, 68, 39, 28, 8, 5, 2, 1, 1;
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MAPLE
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b:= proc(u, o) option remember;
`if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
add(expand(b(u+j-1, o-j)*x^(o-j)), j=1..o))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);
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MATHEMATICA
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b[u_, o_] := b[u, o] = If[u + o == 0, 1, Sum[b[u - j, o + j - 1], {j, 1, u}] + Sum[Expand[b[u + j - 1, o - j]*x^(o - j)], {j, 1, o}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0] ]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A000110, A092923, A264451, A264452, A264453, A264454, A264455, A264456, A264457, A264458, A264459.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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