A317588 - OEIS

%I #6 Aug 01 2018 09:23:01

%S 1,1,2,3,4,3,6,3,5,6,7,5,8,5,7,10,7,6,12,7,12,14,10,11,18,11,13,16,18,

%T 15,35,16,26,24,27,26,47,33,44,48,58,48,76,63,81,79,98,94,123,109,135,

%U 131,148,140,162,149,152,162,166,175,202,191,221,232,233

%N Number of uniformly normal integer partitions of n.

%C An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

%H A. David Christopher and M. Davamani Christober, <a href="http://emis.impa.br/EMIS/journals/GMN/yahoo_site_admin/assets/docs/1_GMN-2492-V13N2.77213831.pdf">Relatively Prime Uniform Partitions</a>, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp.1-12.

%e The a(6) = 6 uniformly normal integer partitions are (6), (33), (321), (222), (2211), (111111). Missing from this list are (51), (42), (411), (3111), (21111).

%e The a(21) = 14 uniformly normal integer partitions (n = 21):

%e   (n),

%e   (777),

%e   (654321),

%e   (4443321), (3333333),

%e   (44432211), (44333211), (44332221),

%e   (4432221111), (4333221111), (4332222111),

%e   (433322211),

%e   (22222221111111),

%e   (111111111111111111111).

%t uninrmQ[q_]:=Or[q=={}||Length[Union[q]]==1,And[Union[q]==Range[Max[q]],uninrmQ[Sort[Length/@Split[q],Greater]]]];

%t Table[Length[Select[IntegerPartitions[n],uninrmQ]],{n,0,30}]

%Y Cf. A181819, A182850, A275870, A317081, A317088, A317246, A317246, A317491, A317589, A317590.

%K nonn

%O 1,3

%A _Gus Wiseman_, Aug 01 2018