A230797 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, up, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-3)/2)), read by rows.
1, 1, 2, 6, 24, 104, 16, 528, 192, 3296, 1472, 272, 23168, 12800, 4352, 179712, 132352, 42880, 7936, 1573632, 1366016, 530432, 158720, 15207424, 14781952, 7662336, 1911296, 353792, 158880768, 178102272, 101713920, 31813632, 8491008, 1801996288, 2282645504
Offset: 0
Examples
T(5,1) = 16: 13254, 14253, 14352, 15243, 15342, 23154, 24153, 24351, 25143, 25341, 34152, 34251, 35142, 35241, 45132, 45231. T(7,2) = 272: 1325476, 1326475, 1326574, ..., 6735241, 6745132, 6745231. Triangle T(n,k) begins: : 0 : 1; : 1 : 1; : 2 : 2; : 3 : 6; : 4 : 24; : 5 : 104, 16; : 6 : 528, 192; : 7 : 3296, 1472, 272; : 8 : 23168, 12800, 4352; : 9 : 179712, 132352, 42880, 7936; : 10 : 1573632, 1366016, 530432, 158720;
Links
- Alois P. Heinz, Rows n = 0..170, flattened
Crossrefs
Programs
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Maple
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand( add(b(u-j, o+j-1, [1, 3, 1, 3][t])*`if`(t=4, x, 1), j=1..u)+ add(b(u+j-1, o-j, [2, 2, 4, 2][t]), 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); # Alois P. Heinz, Oct 30 2013 -
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, 3}[[t]]]*If[t == 4, x, 1], {j, 1, u}] + Sum[b[u+j-1, o-j, {2, 2, 4, 2}[[t]]], {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, Oct 24 2016, after Alois P. Heinz *)