proposed

approved

editing

proposed

antinrmQ[ptn_]:=And[!normQ[ptn], If[&&(Length[ptn]==1, True, ||antinrmQ[Sort[Length/@Split[ptn]]]]]);

Cf. A181819, A275870, A305563, A317088, A317245 (supernormal), , A317491 (fully normal), , A317589 (uniformly normal), , A319149, A319810, A325372.

allocated for Gus WisemanNumber of totally abnormal integer partitions of n.

0, 0, 1, 1, 2, 1, 3, 1, 4, 2, 5, 1, 8, 1, 7, 5, 10, 2, 16, 4, 21, 15, 24, 17, 49, 29, 53, 53, 84, 65, 121, 92, 148, 141, 186, 179, 280, 223, 317, 318, 428, 387, 576, 512, 700, 734, 899, 900, 1260, 1207, 1551, 1668, 2041, 2109, 2748, 2795, 3463, 3775, 4446

0,5

A multiset is normal if its union is an initial interval of positive integers. A multiset is totally abnormal if it is not normal and either it is a singleton or its multiplicities form a totally abnormal multiset.

The Heinz numbers of these partitions are given by A325372.

The a(2) = 1 through a(12) = 8 totally abnormal partitions (A = 10, B = 11, C = 12):

(2) (3) (4) (5) (6) (7) (8) (9) (A) (B) (C)

(22) (33) (44) (333) (55) (66)

(222) (2222) (3322) (444)

(3311) (4411) (3333)

(22222) (4422)

(5511)

(222222)

(333111)

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];

antinrmQ[ptn_]:=And[!normQ[ptn], If[Length[ptn]==1, True, antinrmQ[Sort[Length/@Split[ptn]]]]];

Table[Length[Select[IntegerPartitions[n], antinrmQ]], {n, 0, 30}]

Cf. A181819, A275870, A305563, A317088, A317245 (supernormal), A317491 (fully normal), A317589 (uniformly normal), A319149, A319810, A325372.

allocated

nonn

Gus Wiseman, May 01 2019

approved

editing

allocated for Gus Wiseman

allocated

approved