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A227884
Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, up; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2)-1), read by rows.
5
1, 1, 2, 6, 19, 5, 70, 50, 331, 328, 61, 1863, 2154, 1023, 11637, 16751, 10547, 1385, 81110, 144840, 102030, 34900, 635550, 1314149, 1109973, 518607, 50521, 5495339, 12735722, 13046040, 6858598, 1781101, 51590494, 134159743, 157195762, 97348436, 36004400
OFFSET
0,3
LINKS
EXAMPLE
T(4,1) = 5: 1324, 1423, 2314, 2413, 3412.
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 6;
: 4 : 19, 5;
: 5 : 70, 50;
: 6 : 331, 328, 61;
: 7 : 1863, 2154, 1023;
: 8 : 11637, 16751, 10547, 1385;
: 9 : 81110, 144840, 102030, 34900;
: 10 : 635550, 1314149, 1109973, 518607, 50521;
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)+
add(b(u+j-1, o-j, 2)*`if`(t=3, x, 1), j=1..o)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..15);
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = If[u+o==0, 1, Expand[Sum[b[u-j, o+j-1, {1, 3, 1}[[t]]], {j, 1, u}]+Sum[b[u+j-1, o-j, 2]*If[t==3, x, 1], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0, 1]];
Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Mar 29 2017, translated from Maple *)
CROSSREFS
Columns k=0-1 give: A177477, A227883.
T(2n,n-1) gives A000364(n) for n>=2.
Row sums give: A000142.
Sequence in context: A262971 A253380 A128123 * A186769 A213400 A282080
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Oct 25 2013
STATUS
approved