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A260670
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Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 23-1; triangle T(n,k), n>=0, 0<=k<=A125811(n)-1, read by rows.
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13
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1, 1, 2, 5, 1, 15, 6, 3, 52, 32, 23, 10, 3, 203, 171, 152, 98, 62, 22, 11, 1, 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4, 4140, 5444, 6460, 6082, 5513, 4302, 3328, 2141, 1393, 780, 432, 187, 88, 24, 6, 21147, 32919, 43626, 46508, 46880, 41979, 36774
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OFFSET
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0,3
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COMMENTS
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Patterns 1-32, 3-12, 21-3 give the same sequence.
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LINKS
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FORMULA
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EXAMPLE
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T(3,1) = 1: 231.
T(4,1) = 6: 1342, 2314, 2413, 2431, 3241, 4231.
T(4,2) = 3: 2341, 3412, 3421.
T(5,2) = 23: 13452, 14523, 14532, 23415, 23514, 23541, 24351, 25341, 32451, 34125, 34152, 34215, 35124, 35142, 35214, 35412, 35421, 42351, 43512, 43521, 52341, 53412, 53421.
T(5,3) = 10: 23451, 24513, 24531, 34251, 35241, 45123, 45132, 45213, 45312, 45321.
T(5,4) = 3: 34512, 34521, 45231.
Triangle T(n,k) begins:
0 : 1;
1 : 1;
2 : 2;
3 : 5, 1;
4 : 15, 6, 3;
5 : 52, 32, 23, 10, 3;
6 : 203, 171, 152, 98, 62, 22, 11, 1;
7 : 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4;
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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^u), 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^u], {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, Jan 16 2017, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A000110, A264460, A264461, A264462, A264463, A264464, A264465, A264466, A264467, A264468, A264469.
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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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