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 A386584 #5 Aug 05 2025 22:55:17
%S A386584 0,0,0,0,0,1,0,0,0,1,0,0,1,0,1,0,0,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,2,1,
%T A386584 1,1,0,0,1,0,2,1,2,1,1,0,0,0,1,2,2,2,2,1,1,0,0,1,0,3,2,4,2,2,1,1,0,0,
%U A386584 0,0,3,2,4,3,3,2,1,1
%N A386584 Triangle read by rows where T(n,k) is the number of length k>=0 integer partitions of n having no permutation without any adjacent equal parts (inseparable).
%C A386584 A multiset is inseparable iff it has no anti-run permutations, where an anti-run is a sequence without any adjacent equal parts. Inseparable partitions (A325535) are different from partitions of inseparable type (A386586).
%F A386584 T(n,k) = A072233(n,k) - A386583(n,k).
%e A386584 Row n = 10 counts the following partitions:
%e A386584 . . 55 . 7111 61111 511111 4111111 31111111 211111111 1111111111
%e A386584 4222 22222 421111 3211111 22111111
%e A386584 3331 331111
%e A386584 222211
%e A386584 Triangle begins:
%e A386584 0
%e A386584 0 0
%e A386584 0 0 1
%e A386584 0 0 0 1
%e A386584 0 0 1 0 1
%e A386584 0 0 0 0 1 1
%e A386584 0 0 1 1 1 1 1
%e A386584 0 0 0 0 2 1 1 1
%e A386584 0 0 1 0 2 1 2 1 1
%e A386584 0 0 0 1 2 2 2 2 1 1
%e A386584 0 0 1 0 3 2 4 2 2 1 1
%e A386584 0 0 0 0 3 2 4 3 3 2 1 1
%e A386584 0 0 1 1 3 2 6 4 4 3 2 1 1
%e A386584 0 0 0 0 4 3 6 5 6 4 3 2 1 1
%e A386584 0 0 1 0 4 3 9 6 8 5 5 3 2 1 1
%e A386584 0 0 0 1 4 3 9 7 10 8 6 5 3 2 1 1
%e A386584 0 0 1 0 5 3 12 8 13 9 10 6 5 3 2 1 1
%e A386584 0 0 0 0 5 4 12 10 16 12 12 9 7 5 3 2 1 1
%e A386584 0 0 1 1 5 4 16 11 20 15 17 12 10 7 5 3 2 1 1
%e A386584 0 0 0 0 6 4 16 13 24 18 21 16 14 10 7 5 3 2 1 1
%e A386584 0 0 1 0 6 4 20 14 29 21 28 20 19 13 11 7 5 3 2 1 1
%t A386584 insepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]=={};
%t A386584 Table[Length[Select[IntegerPartitions[n,{k}],insepQ]],{n,0,15},{k,0,n}]
%Y A386584 Inseparable case of A008284 or A072233.
%Y A386584 Row sums are A325535, ranked by A335448.
%Y A386584 For separable instead of inseparable we have A386583, sums A325534, ranks A335433.
%Y A386584 For separable type we have A386585, sums A336106, ranks A335127.
%Y A386584 For inseparable type we have A386586, sums A025065, ranks A335126.
%Y A386584 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386584 A124762 gives inseparability of standard compositions, separability A333382.
%Y A386584 A336103 counts normal separable multisets, inseparable A336102.
%Y A386584 A386633 counts separable set partitions, row sums of A386635.
%Y A386584 A386634 counts inseparable set partitions, row sums of A386636.
%Y A386584 Cf. A106351, A111133, A238130, A239455, A335434, A351293, A386575, A386576, A386580, A386581, A386577.
%K A386584 nonn,tabl
%O A386584 0,33
%A A386584 _Gus Wiseman_, Aug 05 2025