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A304406 Number of partitions of n in which the sequence of the sum of the same summands is nonincreasing.

%I A304406 #25 Oct 21 2022 22:13:21
%S A304406 1,1,2,2,4,3,6,5,9,8,11,11,20,16,20,21,32,30,41,38,50,48,62,64,89,81,
%T A304406 97,100,123,123,151,154,187,183,221,221,279,272,312,316,377,376,446,
%U A304406 460,531,547,628,641,754,746,841,856,990,1007,1145,1167,1325,1346,1519,1567,1776
%N A304406 Number of partitions of n in which the sequence of the sum of the same summands is nonincreasing.
%C A304406 Number of integer partitions of n with weakly increasing run-sums. - _Gus Wiseman_, Oct 21 2022
%e A304406 n |                      | Sequence of the sum of the same summands
%e A304406 --+----------------------+-----------------------------------------
%e A304406 1 | 1                    | 1
%e A304406 2 | 2                    | 2
%e A304406   | 1+1                  | 2
%e A304406 3 | 3                    | 3
%e A304406   | 1+1+1                | 3
%e A304406 4 | 4                    | 4
%e A304406   | 2+2                  | 4
%e A304406   | 2+1+1                | 2, 2
%e A304406   | 1+1+1+1              | 4
%e A304406 5 | 5                    | 5
%e A304406   | 2+1+1+1              | 3, 2
%e A304406   | 1+1+1+1+1            | 5
%e A304406 6 | 6                    | 6
%e A304406   | 3+3                  | 6
%e A304406   | 3+1+1+1              | 3, 3
%e A304406   | 2+2+2                | 6
%e A304406   | 2+1+1+1+1            | 4, 2
%e A304406   | 1+1+1+1+1+1          | 6
%t A304406 Table[Length[Select[IntegerPartitions[n],LessEqual@@Total/@Split[#]&]],{n,0,30}] (* _Gus Wiseman_, Oct 21 2022 *)
%Y A304406 Cf. A100882.
%Y A304406 These partitions are ranked by A357861.
%Y A304406 The complement is A357865, ranked by A357850.
%Y A304406 The opposite version is A304405, ranked by A357875.
%Y A304406 The strict version is A304430, ranked by A357864.
%Y A304406 The strict opposite version is A304428, ranked by A357862.
%Y A304406 Number of rows in A354584 summing to n that are weakly decreasing.
%Y A304406 A000041 counts integer partitions, strict A000009.
%Y A304406 A304442 counts partitions with equal run-sums, distinct A353837.
%Y A304406 Cf. A047966, A087980, A098859, A239312, A275870, A353832, A353833, A353864.
%K A304406 nonn
%O A304406 0,3
%A A304406 _Seiichi Manyama_, May 12 2018