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A386583 Triangle read by rows where T(n,k) is the number of length k integer partitions of n having a permutation without any adjacent equal parts (separable).

%I A386583 #23 Aug 05 2025 21:34:06
%S A386583 1,0,1,0,1,0,0,1,1,0,0,1,1,1,0,0,1,2,2,0,0,0,1,2,2,1,0,0,0,1,3,4,1,1,
%T A386583 0,0,0,1,3,5,3,2,0,0,0,0,1,4,6,4,3,1,0,0,0,0,1,4,8,6,5,1,1,0,0,0,0,1,
%U A386583 5,10,8,8,3,2,0,0,0,0,0,1,5,11,12,11,5,3,1,0,0,0,0
%N A386583 Triangle read by rows where T(n,k) is the number of length k integer partitions of n having a permutation without any adjacent equal parts (separable).
%C A386583 A multiset is separable iff it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
%C A386583 Separable partitions (A325534) are different from partitions of separable type (A386585).
%C A386583 Are the rows all unimodal?
%C A386583 Some rows are not unimodal: T(200, k=26..30) = 149371873744, 153304102463, 152360653274, 152412869411, 147228477998. - _Alois P. Heinz_, Aug 04 2025
%H A386583 Alois P. Heinz, <a href="/A386583/b386583.txt">Rows n = 0..200, flattened</a>
%e A386583 Row n = 9 counts the following partitions:
%e A386583   (9)  (5,4)  (4,3,2)  (3,3,2,1)  (3,2,2,1,1)  (2,2,2,1,1,1)
%e A386583        (6,3)  (4,4,1)  (4,2,2,1)  (3,3,1,1,1)
%e A386583        (7,2)  (5,2,2)  (4,3,1,1)  (4,2,1,1,1)
%e A386583        (8,1)  (5,3,1)  (5,2,1,1)
%e A386583               (6,2,1)
%e A386583               (7,1,1)
%e A386583 Triangle begins:
%e A386583   1
%e A386583   0  1
%e A386583   0  1  0
%e A386583   0  1  1  0
%e A386583   0  1  1  1  0
%e A386583   0  1  2  2  0  0
%e A386583   0  1  2  2  1  0  0
%e A386583   0  1  3  4  1  1  0  0
%e A386583   0  1  3  5  3  2  0  0  0
%e A386583   0  1  4  6  4  3  1  0  0  0
%e A386583   0  1  4  8  6  5  1  1  0  0  0
%e A386583   0  1  5 10  8  8  3  2  0  0  0  0
%e A386583   0  1  5 11 12 11  5  3  1  0  0  0  0
%e A386583   0  1  6 14 14 15  8  6  1  1  0  0  0  0
%e A386583   0  1  6 16 19 20 11  9  3  2  0  0  0  0  0
%e A386583   0  1  7 18 23 27 17 14  5  3  1  0  0  0  0  0
%e A386583   0  1  7 21 29 34 23 20  9  6  1  1  0  0  0  0  0
%e A386583   0  1  8 24 34 43 32 28 13 10  3  2  0  0  0  0  0  0
%e A386583   0  1  8 26 42 53 42 38 20 15  5  3  1  0  0  0  0  0  0
%e A386583   0  1  9 30 48 66 55 52 28 23  9  6  1  1  0  0  0  0  0  0
%e A386583   0  1  9 33 58 80 70 68 41 33 14 10  3  2  0  0  0  0  0  0  0
%e A386583   ...
%t A386583 sepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]!={};
%t A386583 Table[Length[Select[IntegerPartitions[n,{k}],sepQ]],{n,0,15},{k,0,n}]
%Y A386583 Separable case of A008284.
%Y A386583 Row sums are A325534, ranked by A335433.
%Y A386583 For inseparable instead separable we have A386584, sums A325535, ranks A335448.
%Y A386583 For separable type instead of separable we have A386585, sums A336106, ranks A335127.
%Y A386583 For inseparable type instead of separable we have A386586, sums A025065, ranks A335126.
%Y A386583 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386583 A124762 gives inseparability of standard compositions, separability A333382.
%Y A386583 A239455 counts Look-and-Say partitions, ranks A351294.
%Y A386583 A336103 counts normal separable multisets, inseparable A336102.
%Y A386583 A351293 counts non-Look-and-Say partitions, ranks A351295.
%Y A386583 A386633 counts separable set partitions, row sums of A386635.
%Y A386583 A386634 counts inseparable set partitions, row sums of A386636.
%Y A386583 Cf. A106351, A111133, A238130, A335434, A386575, A386576, A386580, A386581, A386577.
%K A386583 nonn,tabl
%O A386583 0,18
%A A386583 _Gus Wiseman_, Aug 03 2025