A317328 Total number of distinct run lengths summed over all permutations of [n] with exactly k distinct lengths of increasing runs.

0, 1, 2, 10, 41, 238, 1560, 11822, 99884, 944951, 9851550, 112297557, 1389182296, 18548151140, 265854443194, 4071965025912, 66379311227650, 1147606112273168, 20974771814051529, 404111977603746584, 8186206878478673819, 173948483818885099616, 3868878742247603336434, 89892706223315981903289

Offset: 0

Author

Alois P. Heinz, Jul 25 2018

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..60

Crossrefs

Cf. A317327.

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:
a:= n-> (p-> add(coeff(p, x, i)*i, i=0..degree(p)))(b(n, 0$2, {})):
seq(a(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}]];
a[n_] := Function[p, Sum[Coefficient[p, x, i]*i, {i, 0, Exponent[p, x]}]][
     b[n, 0, 0, {}]];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 15 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..floor((sqrt(1+8*n)-1)/2)} k * A317327(n,k).