A336771 Total sum of the left-to-right maxima in all compositions of n into distinct parts.

0, 1, 2, 8, 11, 22, 53, 75, 123, 193, 418, 538, 894, 1268, 1950, 3567, 4799, 7143, 10355, 14968, 20701, 36398, 46420, 69071, 94972, 136385, 182522, 259104, 402405, 527090, 741569, 1015491, 1397661, 1880541, 2567202, 3392612, 5153156, 6553844, 9088372, 12040797

Offset: 0

Author

Alois P. Heinz, Aug 04 2020

Keywords

• nonn

Examples

			a(6) = 53 = 6+4+5+5+3+3+6+4+6+5+6: (1)(2)(3), (1)(3)2, (2)1(3), (2)(3)1, (3)12, (3)21, (2)(4), (4)2, (1)(5), (5)1, (6).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000009, A000254, A003056, A032020, A336511, A336718, A336770.

Programs

Maple

b:= proc(n, i, k, m) option remember; `if`(i
      (2*i-k+1)*k/2, 0, `if`(n=0, [1, 0], b(n, i-1, k, m)+
      (p-> p+[0, p[1]*i/(m+1-k)])(b(n-i, min(n-i, i-1), k-1, m))))
    end:
a:= n-> add(b(n$2, k$2)[2]*k!, k=1..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);

Mathematica

b[n_, i_, k_, m_] := b[n, i, k, m] = If[i < k || n >
    (2*i - k + 1)*k/2, {0, 0}, If[n == 0, {1, 0}, b[n, i - 1, k, m] +
    Function[p, p+{0, p[[1]]*i/(m+1-k)}][b[n-i, Min[n-i, i-1], k-1, m]]]];
a[n_] := Sum[b[n, n, k, k][[2]]*k!, {k, 1, Floor[(Sqrt[8*n + 1] - 1)/2]}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)