A345166 Number of separable integer partitions of n without an alternating permutation.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 5, 6, 7, 10, 14, 18, 21, 27, 35, 42, 54, 65, 78, 95, 117, 140, 170, 202, 239, 286, 343, 401, 476, 562, 660, 775, 910, 1056, 1241, 1444, 1678, 1948, 2267, 2615, 3031, 3502, 4036, 4647, 5356, 6143, 7068, 8101, 9274, 10613, 12151, 13856

Offset: 0

Author

Gus Wiseman, Jun 13 2021

Keywords

• nonn

Comments

A partition is separable if it has an anti-run permutation (no adjacent parts equal).

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

The partitions counted by this sequence are those with 2m-1 parts with m being the multiplicity of a part which is neither the smallest or largest part. For example, 4322221 is such a partition since the multiplicity of 2 is 4, the total number of parts is 7, and 2 is neither the smallest or largest part. - Andrew Howroyd, Jan 15 2024

Examples

			The a(10) = 1 through a(16) = 6 partitions:
    32221  42221  52221  62221    43331    43332    53332
                         3222211  72221    53331    63331
                                  4222211  82221    92221
                                           3322221  4322221
                                           5222211  6222211
                                                    322222111
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Allowing alternating permutations gives A325534, ranked by A335433.

Not requiring separability gives A345165, ranked by A345171.

Permutations of this type are ranked by A345169.

The Heinz numbers of these partitions are A345173.

Numbers with a factorization of this type are A348609.

A000041 counts integer partitions.

A001250 counts alternating permutations, complement A348615.

A003242 counts anti-run compositions.

A005649 counts anti-run patterns.

A025047 counts alternating or wiggly compositions, also A025048, A025049.

A325535 counts inseparable partitions, ranked by A335448.

A344654 counts non-twin partitions w/o alt permutation, rank A344653.

A345162 counts normal partitions w/o alt permutation, complement A345163.

A345170 counts partitions w/ alt permutation, ranked by A345172.

Cf. A000070, A103919, A335126, A344604, A344607, A344615, A344740, A344742, A345164, A345166, A345168, A345192, A348379.

Programs

Mathematica

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!MatchQ[#,{_,x_,x_,_}]&]!={}&&Select[Permutations[#],wigQ]=={}&]],{n,0,15}]

Formula

The Heinz numbers of these partitions are A345173 = A345171 /\ A335433.

a(n) = A325534(n) - A345170(n). - Andrew Howroyd, Jan 15 2024

Extensions

a(26) onwards from Andrew Howroyd, Jan 15 2024