cp's OEIS Frontend

A332277 Number of widely totally normal integer partitions of n.

%I A332277 #12 Jun 26 2020 06:19:19
%S A332277 1,1,1,2,2,2,2,2,2,3,4,4,2,4,4,6,3,5,7,6,8,12,9,12,13,11,12,18,17,12,
%T A332277 32,19,25,33,30,28,44,33,43,57,51,60,83,70,83,103,96,97,125,117,134,
%U A332277 157,157,171,226,215,238,278,302,312,359,357,396,450,444,477,580
%N A332277 Number of widely totally normal integer partitions of n.
%C A332277 A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.
%C A332277 Also the number of widely totally normal reversed integer partitions of n.
%e A332277 The a(n) partitions for n = 1, 4, 10, 11, 16, 18:
%e A332277   1  211   4321        33221        443221            543321
%e A332277      1111  33211       322211       4432111           4333221
%e A332277            322111      332111       1111111111111111  4432221
%e A332277            1111111111  11111111111                    4433211
%e A332277                                                       43322211
%e A332277                                                       44322111
%e A332277                                                       111111111111111111
%t A332277 recnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],recnQ[Length/@Split[ptn]]]];
%t A332277 Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]
%Y A332277 Normal partitions are A000009.
%Y A332277 Taking multiplicities instead of run-lengths gives A317245.
%Y A332277 Constantly recursively normal partitions are A332272.
%Y A332277 The Heinz numbers of these partitions are A332276.
%Y A332277 The case of all compositions (not just partitions) is A332279.
%Y A332277 The co-strong version is A332278.
%Y A332277 The recursive version is A332295.
%Y A332277 The narrow version is a(n) + 1 for n > 1.
%Y A332277 Cf. A181819, A316496, A317081, A317256, A317491, A317588, A329746, A329747, A332289, A332290, A332291, A332296, A332297, A332336, A332337, A332340.
%K A332277 nonn
%O A332277 0,4
%A A332277 _Gus Wiseman_, Feb 12 2020
%E A332277 a(61)-a(66) from _Jinyuan Wang_, Jun 26 2020