A361651 Number T(n,k) of permutations p of [n] such that p(i), p(i+k), p(i+2k),... form an up-down sequence for i in [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 0, 1, 2, 0, 2, 3, 6, 0, 5, 6, 12, 24, 0, 16, 20, 30, 60, 120, 0, 61, 80, 90, 180, 360, 720, 0, 272, 350, 420, 630, 1260, 2520, 5040, 0, 1385, 1750, 2240, 2520, 5040, 10080, 20160, 40320, 0, 7936, 10080, 13440, 15120, 22680, 45360, 90720, 181440, 362880
Offset: 0
Examples
Triangle T(n,k) begins: 1; 0, 1; 0, 1, 2; 0, 2, 3, 6; 0, 5, 6, 12, 24; 0, 16, 20, 30, 60, 120; 0, 61, 80, 90, 180, 360, 720; 0, 272, 350, 420, 630, 1260, 2520, 5040; 0, 1385, 1750, 2240, 2520, 5040, 10080, 20160, 40320; ...
Links
- Alois P. Heinz, Rows n = 0..150, flattened
Crossrefs
Programs
-
Maple
b:= proc(u, o) option remember; `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u)) end: T:= (n, k)-> `if`(n=0, 1, `if`(k=0, 0, (l-> mul(b(s, 0), s=l)* combinat[multinomial](n, l[]))([floor((n+i)/k)$i=0..k-1]))): seq(seq(T(n, k), k=0..n), n=0..10); -
Mathematica
multinomial[n_, k_List] := n!/Times @@ (k!); b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[o-1+j, u-j], {j, 1, u}]]; T[n_, k_] := If[n == 0, 1, If[k == 0, 0, Function[l, Product[b[s, 0], {s, l}]*multinomial[n, l]][Table[Floor[(n+i)/k], {i, 0, k-1}]]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Nov 22 2023, after Alois P. Heinz *)
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