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

0,3

Links

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

Formula

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

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 *)

Crossrefs

Cf. A317327.

Sequence in context: A297047 A037561 A135512 * A368678 A083937 A085224

Adjacent sequences: A317325 A317326 A317327 * A317329 A317330 A317331

Keyword

nonn

Author

Alois P. Heinz, Jul 25 2018

Status

approved