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Cf. A035470, A064914, A305551, A336137, A365543, A365663, A365661, A365663, A366320, A365925, A367094, A371788.

The strict complement is A237258, ranks A357854.

This is the "bi-" version of A321451, ranks A321453, complement A321452, ranks A321454.

The complement is the "bi-" version of A321452, ranks A321454.

The strict case is A371794 (, bisections A321142, A078408), complement A237258 (ranks A357854).

Cf. `A035470, A064914, ~A299701, ~A300061, A305551, ~A317583, A318434, ~A321455, ~A322794, `A326518, `A326534, ~A336137, ~A357879, A365543, A365663, A365661, `A366320, `A365381, `A365925, A367094, ~A371732, ~A371733, ~A371735, `A371788.

allocated for Gus WisemanNumber of non-biquanimous integer partitions of n.

0, 1, 1, 3, 2, 7, 5, 15, 8, 30, 17, 56, 24, 101, 46, 176, 64, 297, 107, 490, 147, 792, 242, 1255, 302, 1958, 488, 3010, 629, 4565, 922

0,4

A finite multiset of numbers is defined to be biquanimous iff it can be partitioned into two multisets with equal sums. Biquanimous partitions are counted by A002219 and ranked by A357976.

The a(1) = 1 through a(8) = 8 partitions:

(1) (2) (3) (4) (5) (6) (7) (8)

(21) (31) (32) (42) (43) (53)

(111) (41) (51) (52) (62)

(221) (222) (61) (71)

(311) (411) (322) (332)

(2111) (331) (521)

(11111) (421) (611)

(511) (5111)

(2221)

(3211)

(4111)

(22111)

(31111)

(211111)

(1111111)

biqQ[y_]:=MemberQ[Total/@Subsets[y], Total[y]/2];

Table[Length[Select[IntegerPartitions[n], Not@*biqQ]], {n, 0, 15}]

The complement is counted by A002219 aerated, ranks A357976.

Even bisection is A006827, odd A058695.

This is the "bi-" version of A321451, ranks A321453, complement A321452, ranks A321454.

These partitions have ranks A371731.

The strict case is A371794 (bisections A321142, A078408), complement A237258 (ranks A357854).

A108917 counts knapsack partitions, ranks A299702, strict A275972.

A366754 counts non-knapsack partitions, ranks A299729, strict A316402.

A371736 counts non-quanimous strict partitons, complement A371737.

A371781 lists numbers with biquanimous prime signature, complement A371782.

A371783 counts k-quanimous partitions.

A371789 counts non-quanimous sets, differences A371790.

A371791 counts biquanimous sets, differences A232466.

A371792 counts non-biquanimous sets, differences A371793.

A371796 counts quanimous sets, differences A371797.

Cf. `A035470, A064914, ~A299701, ~A300061, A305551, ~A317583, A318434, ~A321455, ~A322794, `A326518, `A326534, ~A336137, ~A357879, A365543, A365663, A365661, `A366320, `A365381, `A365925, A367094, ~A371732, ~A371733, ~A371735, `A371788.

allocated

nonn,more

Gus Wiseman, Apr 07 2024

approved

editing

allocated for Gus Wiseman

allocated

approved