A366883 a(n) = greatest number of unrestricted partitions between successive strict partitions of n, with partitions listed in Mathematica order.

0, 0, 0, 2, 1, 2, 4, 7, 5, 7, 13, 19, 26, 22, 27, 45, 61, 78, 99, 87, 107, 166, 213, 267, 324, 396, 361, 434, 644, 806, 975, 1167, 1385, 1641, 1541, 1819, 2611, 3200, 3786, 4451, 5196, 6054, 7025, 6744, 7823, 10983, 13190, 15378, 17777, 20482, 23534, 26978, 30860, 30094, 34422, 47456, 56127, 64558, 73684, 83887, 95277, 108061, 122309, 138265, 136498

Offset

3,4

Comments

For Mathematica order of partitions, see A080577.

Links

Table of n, a(n) for n=3..67.

Example

The 6 strict partitions of 8 are [8], [7,1], [6,2], [5,3], [5,2,1], [4,3,1]. These are in positions 1,2,3,5,6,9 among the 22 unrestricted partitions of 8, with [5,1,1,1] and [4,4] positioned between [5,2,1] and [4,3,1]. Thus, a(8) = 2.

Mathematica

q[n_, k_] := q[n, k] = Select[IntegerPartitions[n], DeleteDuplicates[#] == # &][[k]];
u[n_, k_] := u[n, k] = Position[IntegerPartitions[n], q[n, k]];
t[n_] := Table[u[n, k], {k, 1, PartitionsQ[n]}];
Table[-1 + Max[Differences[t[n]]], {n, 3, 10}]

Crossrefs

Cf. A000009 (strict partitions), A000041 (partitions), A080577.

Sequence in context: A058553 A038067 A136102 * A306810 A325747 A325672

Adjacent sequences: A366880 A366881 A366882 * A366884 A366885 A366886

Keyword

nonn

Author

Clark Kimberling, Dec 23 2023

Extensions

Terms a(41) onward from Max Alekseyev, Oct 10 2024

Status

approved