A364808 a(n) = sum of minimal runlengths of all the partitions of n.

1, 3, 5, 9, 11, 20, 22, 36, 44, 63, 74, 114, 128, 180, 224, 298, 355, 485, 573, 760, 922, 1174, 1419, 1836, 2189, 2756, 3341, 4160, 4988, 6217, 7412, 9131, 10941, 13326, 15916, 19379, 22988, 27770, 33017, 39662, 46919, 56223, 66308, 79047, 93187, 110512

Offset: 1

Author

Clark Kimberling, Sep 10 2023

Keywords

• nonn

Examples

			The partitions of 4 are [4], [3,1], [2,2], [2,1,1], [1,1,1,1], with runlengths {1}, {1,1}, {2}, {1,2}, {4} having minima 1, 1, 2, 1, 4, with sum 9, so that a(4) = 9.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000041, A264397 (sum of maximal runlengths).

Programs

Maple

b:= proc(n, i, m) option remember; `if`(n=0, m, `if`(i=1, min(m, n),
      add(b(n-i*j, i-1, `if`(j=0, m, min(m, j))), j=0..n/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 17 2023

Mathematica

m[n_] := m[n] = Map[Split, IntegerPartitions[n]]
t[n_] := t[n] = Table[Map[Length, m[n][[k]]], {k, 1, PartitionsP[n]}]
Table[Total[Map[Min, t[n]]], {n, 1, 47}]

Python

from sympy.utilities.iterables import partitions
def A364808(n): return sum(min(p.values()) for p in partitions(n)) # Chai Wah Wu, Sep 17 2023