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

%I #27 Oct 22 2022 08:08:28

%S 1,1,2,3,4,6,9,11,14,20,26,33,41,50,64,81,97,120,150,176,210,255,303,

%T 362,426,503,595,703,816,953,1113,1283,1482,1721,1988,2299,2650,3031,

%U 3464,3965,4492,5115,5820,6592,7467,8484,9568,10822,12185,13724,15445,17381,19475,21855

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

%C Number of integer partitions of n with strictly decreasing run-sums. - _Gus Wiseman_, Oct 21 2022

%H Alois P. Heinz, <a href="/A304428/b304428.txt">Table of n, a(n) for n = 0..1000</a> (first 71 terms from Seiichi Manyama)

%F a(n) <= A304405(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 | 2+1 | 1, 2

%e | 1+1+1 | 3

%e 4 | 4 | 4

%e | 3+1 | 1, 3

%e | 2+2 | 4

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

%e 5 | 5 | 5

%e | 4+1 | 1, 4

%e | 3+2 | 2, 3

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

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

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

%e 6 | 6 | 6

%e | 5+1 | 1, 5

%e | 4+2 | 2, 4

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

%e | 3+3 | 6

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

%e | 2+2+2 | 6

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

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

%t Table[Length[Select[IntegerPartitions[n],Greater@@Total/@Split[#]&]],{n,0,30}] (* _Gus Wiseman_, Oct 21 2022 *)

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

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

%Y The opposite version is A304430, ranked by A357864.

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

%Y These partitions are ranked by A357862, complement A357863.

%Y A000041 counts integer partitions, strict A000009.

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

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

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 12 2018