A325712 -id:A325712 - cp's OEIS frontend

A325711 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 2.

1, 1, 2, 3, 5, 5, 8, 9, 10, 12, 14, 15, 17, 19, 19, 23, 23, 26, 26, 30, 29, 33, 32, 39, 35, 40, 39, 45, 41, 50, 44, 53, 49, 54, 51, 64, 53, 61, 59, 70, 59, 74, 62, 75, 71, 75, 68, 91, 72, 85, 79, 90, 77, 98, 83, 101, 89, 96, 86, 120

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 A325534 for a guide to related sequences.

			a(6) counts these 5 partitions:  [6], [5,1], [4,2], [1,4,1], [2,1,2,1].

Cf. A000041, A325334, A325712.

(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]]<=2&]]&,separable]
(* Peter J. C. Moses, May 08 2019 *)

A325718 Number of separable partitions of n in which the number of distinct (repeatable) parts is > 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 10, 19, 29, 50, 74, 112, 158, 225, 311, 424, 564, 751, 979, 1273, 1638, 2082, 2640, 3323, 4159, 5171, 6419, 7895, 9707, 11872, 14481, 17573, 21319, 25709, 30997, 37205, 44619, 53304, 63646, 75698, 89998, 106664, 126306

Offset: 1

Clark Kimberling, May 17 2019

A partition is separable if there is an ordering of its parts in which no consecutive parts are identical. See A325534 for a guide to related sequences.

Cf. A000041, A325534.

(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 *)

a(n) = A325534(n) - A325712(n) for n >= 1.

cp's OEIS Frontend

A325711 Number of separable partitions of n in which the number of distinct (repeatable) parts <= 2.

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