cp's OEIS Frontend

A304428 Number of partitions of n in which the sequence of the sum of the same summands is increasing.

%I A304428 #27 Oct 22 2022 08:08:28
%S A304428 1,1,2,3,4,6,9,11,14,20,26,33,41,50,64,81,97,120,150,176,210,255,303,
%T A304428 362,426,503,595,703,816,953,1113,1283,1482,1721,1988,2299,2650,3031,
%U A304428 3464,3965,4492,5115,5820,6592,7467,8484,9568,10822,12185,13724,15445,17381,19475,21855
%N A304428 Number of partitions of n in which the sequence of the sum of the same summands is increasing.
%C A304428 Number of integer partitions of n with strictly decreasing run-sums. - _Gus Wiseman_, Oct 21 2022
%H A304428 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 A304428 a(n) <= A304405(n).
%e A304428 n |                      | Sequence of the sum of the same summands
%e A304428 --+----------------------+-----------------------------------------
%e A304428 1 | 1                    | 1
%e A304428 2 | 2                    | 2
%e A304428   | 1+1                  | 2
%e A304428 3 | 3                    | 3
%e A304428   | 2+1                  | 1, 2
%e A304428   | 1+1+1                | 3
%e A304428 4 | 4                    | 4
%e A304428   | 3+1                  | 1, 3
%e A304428   | 2+2                  | 4
%e A304428   | 1+1+1+1              | 4
%e A304428 5 | 5                    | 5
%e A304428   | 4+1                  | 1, 4
%e A304428   | 3+2                  | 2, 3
%e A304428   | 3+1+1                | 2, 3
%e A304428   | 2+2+1                | 1, 4
%e A304428   | 1+1+1+1+1            | 5
%e A304428 6 | 6                    | 6
%e A304428   | 5+1                  | 1, 5
%e A304428   | 4+2                  | 2, 4
%e A304428   | 4+1+1                | 2, 4
%e A304428   | 3+3                  | 6
%e A304428   | 3+2+1                | 1, 2, 3
%e A304428   | 2+2+2                | 6
%e A304428   | 2+2+1+1              | 2, 4
%e A304428   | 1+1+1+1+1+1          | 6
%t A304428 Table[Length[Select[IntegerPartitions[n],Greater@@Total/@Split[#]&]],{n,0,30}] (* _Gus Wiseman_, Oct 21 2022 *)
%Y A304428 The weak version is A304405, ranked by A357875.
%Y A304428 The weak opposite version is A304406, ranked by A357861.
%Y A304428 The opposite version is A304430, ranked by A357864.
%Y A304428 Number of rows in A354584 summing to n that are strictly increasing.
%Y A304428 These partitions are ranked by A357862, complement A357863.
%Y A304428 A000041 counts integer partitions, strict A000009.
%Y A304428 A304442 counts partitions with equal run-sums, distinct A353837.
%Y A304428 Cf. A047966, A087980, A098859, A100471, A239312, A275870, A353832, A353864.
%K A304428 nonn
%O A304428 0,3
%A A304428 _Seiichi Manyama_, May 12 2018