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Number of partitions of n in which the sequence of the sum of the same summands is decreasing.
11

%I #18 Oct 22 2022 08:07:47

%S 1,1,2,2,3,3,5,5,6,8,10,10,13,15,18,19,22,26,33,33,38,41,50,53,60,68,

%T 77,84,94,100,116,122,136,148,172,182,206,219,246,258,281,301,341,365,

%U 397,429,466,489,528,572,623,660,728,773,849,895,968,1019,1120,1188,1288

%N Number of partitions of n in which the sequence of the sum of the same summands is decreasing.

%C Number of integer partitions of n with strictly increasing run-sums. - _Gus Wiseman_, Oct 22 2022

%F a(n) <= A304406(n).

%e n | | Sequence of the sum of the same summands

%e --+----------------------+-----------------------------------------

%e 1 | 1 | 1

%e 2 | 2 | 2

%e | 1+1 | 2

%e 3 | 3 | 3

%e | 1+1+1 | 3

%e 4 | 4 | 4

%e | 2+2 | 4

%e | 1+1+1+1 | 4

%e 5 | 5 | 5

%e | 2+1+1+1 | 3, 2

%e | 1+1+1+1+1 | 5

%e 6 | 6 | 6

%e | 3+3 | 6

%e | 2+2+2 | 6

%e | 2+1+1+1+1 | 4, 2

%e | 1+1+1+1+1+1 | 6

%t Table[Length[Select[IntegerPartitions[n],Less@@Total/@Split[#]&]],{n,0,30}] (* _Gus Wiseman_, Oct 22 2022 *)

%Y The weak opposite version is A304405, ranked by A357875.

%Y The weak version is A304406, ranked by A357861.

%Y The opposite version is A304428, ranked by A357862.

%Y Number of rows in A354584 summing to n that are strictly decreasing.

%Y These partitions are ranked by A357864.

%Y A000041 counts integer partitions, strict A000009.

%Y A304442 counts partitions with equal run-sums, distinct A353837.

%Y Cf. A047966, A087980, A098859, A100471, A100881, A239312, A275870, A353832, A353833, A353864.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 12 2018