This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385815 #13 Jul 10 2025 12:53:34
%S A385815 1,0,1,0,2,0,0,3,0,0,0,4,1,0,0,0,5,2,0,0,0,0,7,4,0,0,0,0,0,8,7,0,0,0,
%T A385815 0,0,0,10,12,0,0,0,0,0,0,0,13,16,1,0,0,0,0,0,0,0,15,25,2,0,0,0,0,0,0,
%U A385815 0,0,18,34,4,0,0,0,0,0,0,0,0
%N A385815 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive elements decreasing by 0 or 1.
%e A385815 The partition (8,5,4,2,1) has maximal runs ((8),(5,4),(2,1)) so is counted under T(20,3).
%e A385815 The partition (8,5,3,2,2) has maximal runs ((8),(5),(3,2,2)) so is also counted under T(20,3).
%e A385815 Row n = 9 counts the following partitions:
%e A385815 (9) (6,3) (5,3,1)
%e A385815 (5,4) (7,2)
%e A385815 (3,3,3) (8,1)
%e A385815 (4,3,2) (4,4,1)
%e A385815 (3,2,2,2) (5,2,2)
%e A385815 (3,3,2,1) (6,2,1)
%e A385815 (2,2,2,2,1) (7,1,1)
%e A385815 (3,2,2,1,1) (4,2,2,1)
%e A385815 (2,2,2,1,1,1) (4,3,1,1)
%e A385815 (3,2,1,1,1,1) (5,2,1,1)
%e A385815 (2,2,1,1,1,1,1) (6,1,1,1)
%e A385815 (2,1,1,1,1,1,1,1) (3,3,1,1,1)
%e A385815 (1,1,1,1,1,1,1,1,1) (4,2,1,1,1)
%e A385815 (5,1,1,1,1)
%e A385815 (4,1,1,1,1,1)
%e A385815 (3,1,1,1,1,1,1)
%e A385815 Triangle begins:
%e A385815 1
%e A385815 0 1
%e A385815 0 2 0
%e A385815 0 3 0 0
%e A385815 0 4 1 0 0
%e A385815 0 5 2 0 0 0
%e A385815 0 7 4 0 0 0 0
%e A385815 0 8 7 0 0 0 0 0
%e A385815 0 10 12 0 0 0 0 0 0
%e A385815 0 13 16 1 0 0 0 0 0 0
%e A385815 0 15 25 2 0 0 0 0 0 0 0
%e A385815 0 18 34 4 0 0 0 0 0 0 0 0
%e A385815 0 23 46 8 0 0 0 0 0 0 0 0 0
%e A385815 0 26 62 13 0 0 0 0 0 0 0 0 0 0
%e A385815 0 31 82 22 0 0 0 0 0 0 0 0 0 0 0
%t A385815 Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1<=#2+1&]]==k&]],{n,0,20},{k,0,n}]
%Y A385815 Row sums are A000041, strict A000009.
%Y A385815 Column k = 1 is A034296 (flat or gapless partitions, ranks A066311 or A073491).
%Y A385815 For subsets instead of partitions we have A034839, anti-runs A384893.
%Y A385815 The strict case appears to be A116674.
%Y A385815 For anti-runs instead of runs we have A268193.
%Y A385815 The corresponding rank statistic is A287170.
%Y A385815 For proper runs instead of runs we have A384881.
%Y A385815 For proper anti-runs instead of runs we have A385814.
%Y A385815 A007690 counts partitions with no singletons (ranks A001694), complement A183558.
%Y A385815 A047993 counts partitions with max part = length, rank A106529.
%Y A385815 A098859 counts Wilf partitions, complement A336866 (ranks A325992).
%Y A385815 A116608 counts partitions by distinct parts.
%Y A385815 A116931 counts sparse partitions, ranks A319630.
%Y A385815 Cf. A001227, A008284, A325325, A356226, A384884, A384885, A384887, A384905.
%K A385815 nonn,tabl
%O A385815 0,5
%A A385815 _Gus Wiseman_, Jul 09 2025