A317327 Number T(n,k) of permutations of [n] with exactly k distinct lengths of increasing runs; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
1, 0, 1, 0, 2, 0, 2, 4, 0, 7, 17, 0, 2, 118, 0, 82, 436, 202, 0, 2, 3294, 1744, 0, 1456, 18164, 20700, 0, 1515, 140659, 220706, 0, 50774, 1096994, 2317340, 163692, 0, 2, 10116767, 27136103, 2663928, 0, 3052874, 94670868, 328323746, 52954112, 0, 2, 1021089326, 4317753402, 888178070
Offset: 0
Examples
T(4,1) = 7: 1234, 1324, 1423, 2314, 2413, 3412, 4321. Triangle T(n,k) begins: 1; 0, 1; 0, 2; 0, 2, 4; 0, 7, 17; 0, 2, 118; 0, 82, 436, 202; 0, 2, 3294, 1744; 0, 1456, 18164, 20700; 0, 1515, 140659, 220706; 0, 50774, 1096994, 2317340, 163692; 0, 2, 10116767, 27136103, 2663928; 0, 3052874, 94670868, 328323746, 52954112; ...
Links
- Alois P. Heinz, Rows n = 0..60, flattened
Crossrefs
Programs
-
Maple
b:= proc(u, o, t, s) option remember; `if`(u+o=0, x^(nops(s union {t})-1), add(b(u-j, o+j-1, 1, s union {t}), j=1..u)+ add(b(u+j-1, o-j, t+1, s), j=1..o)) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2, {})): seq(T(n), n=0..16); -
Mathematica
b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, x^(Length[s ~Union~ {t}] - 1), Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}], {j, 1, u}] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]]; T[n_] := With[{p = b[n, 0, 0, {}]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]]; T /@ Range[0, 16] // Flatten (* Jean-François Alcover, Jan 27 2021, after Alois P. Heinz *)