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A263776
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Triangle read by rows: T(n,k) (n>=0, 0<=k<=A002620(n-1)) is the number of permutations of [n] with k nestings.
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15
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1, 1, 2, 5, 1, 14, 8, 2, 42, 45, 25, 7, 1, 132, 220, 198, 112, 44, 12, 2, 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 1, 1430, 4368, 7280, 8400, 7460, 5392, 3262, 1664, 716, 256, 74, 16, 2, 4862, 18564, 38556, 56100, 63648, 59670, 47802, 33338, 20466, 11115
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OFFSET
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0,3
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COMMENTS
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Also the number of permutations of [n] with k crossings (see Corteel, Proposition 4).
Also the number of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 13-2 (alternatively: 2-13, 2-31, or 31-2). - Alois P. Heinz, Nov 14 2015
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
0 : 1;
1 : 1;
2 : 2;
3 : 5, 1;
4 : 14, 8, 2;
5 : 42, 45, 25, 7, 1;
6 : 132, 220, 198, 112, 44, 12, 2;
7 : 429, 1001, 1274, 1092, 700, 352, 140, 42, 9, 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^(j-1)), j=1..o))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
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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^(j-1)], {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 31 2016, after Alois P. Heinz *)
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CROSSREFS
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Columns k=0-10 give: A000108, A002696, A094218, A094219, A120812, A120813, A120814, A120815, A120816, A264496, A264497.
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KEYWORD
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nonn,tabf
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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