A325646 -id:A325646 - cp's OEIS frontend

A325647 Number of separable partitions of n in which the number of distinct (repeatable) parts is 3.

0, 0, 0, 0, 0, 1, 2, 5, 9, 13, 21, 29, 39, 49, 68, 79, 101, 116, 145, 167, 196, 221, 262, 287, 335, 368, 412, 460, 512, 554, 617, 673, 723, 800, 865, 925, 1001, 1090, 1140, 1250, 1317, 1418, 1493, 1619, 1665, 1828, 1884, 2022, 2098, 2275, 2308, 2520, 2564

Offset: 1

Clark Kimberling, May 16 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325646 for a guide to related sequences.

			a(8) counts these 5 partitions:  [5,2,1], [4,3,1], [1,4,1,2], [2,3,2,1], [1,3,1,2,1].

Cf. A000041, A325334, A325646.

(separable=Table[Map[#[[1]]&,Select[Map[{#,Quotient[(1+Length[#]),Max[Map[Length,Split[#]]]]}&,IntegerPartitions[nn]],#[[2]]>1&]],{nn,35}]);
Map[Length[Select[Map[{#,Length[Union[#]]}&,#],#[[2]]==3&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

A325648 Number of separable partitions of n in which the number of distinct (repeatable) parts is 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 10, 19, 28, 48, 69, 102, 138, 190, 254, 331, 420, 534, 662, 818, 1003, 1192, 1438, 1700, 2011, 2334, 2746, 3127, 3625, 4132, 4728, 5311, 6066, 6737, 7624, 8446, 9476, 10437, 11648, 12731, 14133, 15421, 16991, 18461, 20286

Offset: 1

Clark Kimberling, May 16 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325646 for a guide to related sequences.

			a(11) counts these 2 partitions:  [4,1,3,2,1], [3,2,1,2,1,2].

Cf. A000041, A325646.

(separable=Table[Map[#[[1]]&,Select[Map[{#,Quotient[(1+Length[#]),Max[Map[Length,Split[#]]]]}&,IntegerPartitions[nn]],#[[2]]>1&]],{nn,35}]);
Map[Length[Select[Map[{#,Length[Union[#]]}&,#],#[[2]]==4&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

cp's OEIS Frontend

A325647 Number of separable partitions of n in which the number of distinct (repeatable) parts is 3.

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