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 A385814 #10 Jul 10 2025 11:24:19
%S A385814 1,0,1,0,1,1,0,1,1,1,0,2,1,1,1,0,2,2,1,1,1,0,3,2,3,1,1,1,0,3,4,2,3,1,
%T A385814 1,1,0,4,5,4,3,3,1,1,1,0,5,5,6,5,3,3,1,1,1,0,6,8,7,6,6,3,3,1,1,1,0,7,
%U A385814 9,10,8,7,6,3,3,1,1,1
%N A385814 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal proper anti-runs (sequences decreasing by more than 1).
%e A385814 The partition (8,5,4,2,1) has maximal proper anti-runs ((8,5),(4,2),(1)) so is counted under T(20,3).
%e A385814 The partition (8,5,3,2,2) has maximal proper anti-runs ((8,5,3),(2),(2)) so is also counted under T(20,3).
%e A385814 Row n = 8 counts the following partitions:
%e A385814 . 8 611 5111 41111 32111 221111 2111111 11111111
%e A385814 71 521 4211 3221 311111
%e A385814 62 44 332 2222 22211
%e A385814 53 431 3311
%e A385814 422
%e A385814 Triangle begins:
%e A385814 1
%e A385814 0 1
%e A385814 0 1 1
%e A385814 0 1 1 1
%e A385814 0 2 1 1 1
%e A385814 0 2 2 1 1 1
%e A385814 0 3 2 3 1 1 1
%e A385814 0 3 4 2 3 1 1 1
%e A385814 0 4 5 4 3 3 1 1 1
%e A385814 0 5 5 6 5 3 3 1 1 1
%e A385814 0 6 8 7 6 6 3 3 1 1 1
%e A385814 0 7 9 10 8 7 6 3 3 1 1 1
%e A385814 0 9 11 13 12 9 8 6 3 3 1 1 1
%e A385814 0 10 14 16 15 13 10 8 6 3 3 1 1 1
%e A385814 0 12 19 18 21 17 14 11 8 6 3 3 1 1 1
%e A385814 0 14 21 26 23 24 19 15 11 8 6 3 3 1 1 1
%e A385814 0 17 26 31 33 28 26 20 16 11 8 6 3 3 1 1 1
%e A385814 0 19 32 37 40 39 31 28 21 16 11 8 6 3 3 1 1 1
%e A385814 0 23 38 47 50 47 45 34 29 22 16 11 8 6 3 3 1 1 1
%e A385814 0 26 45 57 61 61 54 48 36 30 22 16 11 8 6 3 3 1 1 1
%e A385814 0 31 53 71 75 76 70 60 51 37 31 22 16 11 8 6 3 3 1 1 1
%t A385814 Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1>#2+1&]]==k&]],{n,0,10},{k,0,n}]
%Y A385814 Row sums are A000041, strict A000009.
%Y A385814 Column k = 1 is A003114.
%Y A385814 For anti-runs instead of proper anti-runs we have A268193.
%Y A385814 The corresponding rank statistic is A356228.
%Y A385814 For proper runs instead of proper anti-runs we have A384881.
%Y A385814 For subsets instead of partitions we have A384893, runs A034839.
%Y A385814 The strict case is A384905.
%Y A385814 For runs instead of proper anti-runs we have A385815.
%Y A385814 A007690 counts partitions with no singletons (ranks A001694), complement A183558.
%Y A385814 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A385814 A047993 counts partitions with max part = length, ranks A106529.
%Y A385814 A098859 counts Wilf partitions, complement A336866 (ranks A325992).
%Y A385814 A116608 counts partitions by distinct parts.
%Y A385814 A116931 counts sparse partitions, ranks A319630.
%Y A385814 Cf. A001227, A008284, A089259, A116674, A239455, A325325, A356226, A384880, A384885, A384887, A384906.
%K A385814 nonn,tabl
%O A385814 0,12
%A A385814 _Gus Wiseman_, Jul 09 2025