A389805 The number of partitions of [n] into a set of irreducible balanced subsets.

1, 1, 1, 1, 1, 1, 2, 2, 4, 5, 14, 19, 64, 89, 305, 489, 1947, 3122, 13352, 22777, 111333, 189016, 1026737, 1761266, 11107110, 18557581, 131349930, 218008848, 1773125055, 2850048230, 26118199475, 41383639806, 428054835916, 659845914516, 7587716489188, 11512674828157

Offset

1,7

Comments

The mean of [n]={1..n} is (n+1)/2. A proper subset of [n] is called balanced if its mean value is also (n+1)/2. A balanced subset is referred to as a reducible balanced subset if it contains a smaller balanced subset, otherwise, it is called an irreducible balanced subset.

Partitions of [n] into a set of irreducible balanced subsets are called complete.

Links

Martin Fuller, Table of n, a(n) for n = 1..40

Sean A. Irvine, Java program (github)

Hu Junhua, Balanced samples of the initial segment of natural numbers Chinese Math R & D BBS, Apr 21 2010.

Example

For n = 7, a(7) = 2, since [7] has 2 complete partitions: {{4}, {1, 7}, {2, 6}, {3, 5}} and {{4}, {1, 5, 6}, {2, 3, 7}}.
For n = 8, a(8) = 2, since [8] has 2 complete partitions: {{1, 8}, {2, 7}, {3, 6}, {4, 5}} and {{1, 4, 6, 7}, {2, 3, 5, 8}}.
For n = 9, a(9) = 4, since [9] has 4 complete partitions: {{5}, {1, 9}, {2, 8}, {3, 7}, {4, 6}}, {{5}, {3, 7}, {2, 4, 9}, {1, 6, 8}}, {{5}, {1, 9}, {2, 6, 7}, {3, 4, 8}}, and {{5}, {1, 4, 7, 8}, {2, 3, 6, 9}}.

Crossrefs

Cf. A389806, A389807, A389802.

Sequence in context: A302400 A175954 A371162 * A158649 A357886 A019964

Adjacent sequences: A389802 A389803 A389804 * A389806 A389807 A389809

Keyword

nonn

Author

Hu Junhua, Oct 15 2025

Extensions

a(19)-a(28) from Sean A. Irvine, Oct 21 2025

a(29) onward from Martin Fuller, Jan 27 2026

Status

approved