cp's OEIS Frontend

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

%I A345166 #17 Jan 31 2024 22:23:04
%S A345166 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,
%T A345166 117,140,170,202,239,286,343,401,476,562,660,775,910,1056,1241,1444,
%U A345166 1678,1948,2267,2615,3031,3502,4036,4647,5356,6143,7068,8101,9274,10613,12151,13856
%N A345166 Number of separable integer partitions of n without an alternating permutation.
%C A345166 A partition is separable if it has an anti-run permutation (no adjacent parts equal).
%C A345166 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).
%C A345166 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
%H A345166 Andrew Howroyd, <a href="/A345166/b345166.txt">Table of n, a(n) for n = 0..1000</a>
%F A345166 The Heinz numbers of these partitions are A345173 = A345171 /\ A335433.
%F A345166 a(n) = A325534(n) - A345170(n). - _Andrew Howroyd_, Jan 15 2024
%e A345166 The a(10) = 1 through a(16) = 6 partitions:
%e A345166     32221  42221  52221  62221    43331    43332    53332
%e A345166                          3222211  72221    53331    63331
%e A345166                                   4222211  82221    92221
%e A345166                                            3322221  4322221
%e A345166                                            5222211  6222211
%e A345166                                                     322222111
%t A345166 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A345166 Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!MatchQ[#,{___,x_,x_,___}]&]!={}&&Select[Permutations[#],wigQ]=={}&]],{n,0,15}]
%Y A345166 Allowing alternating permutations gives A325534, ranked by A335433.
%Y A345166 Not requiring separability gives A345165, ranked by A345171.
%Y A345166 Permutations of this type are ranked by A345169.
%Y A345166 The Heinz numbers of these partitions are A345173.
%Y A345166 Numbers with a factorization of this type are A348609.
%Y A345166 A000041 counts integer partitions.
%Y A345166 A001250 counts alternating permutations, complement A348615.
%Y A345166 A003242 counts anti-run compositions.
%Y A345166 A005649 counts anti-run patterns.
%Y A345166 A025047 counts alternating or wiggly compositions, also A025048, A025049.
%Y A345166 A325535 counts inseparable partitions, ranked by A335448.
%Y A345166 A344654 counts non-twin partitions w/o alt permutation, rank A344653.
%Y A345166 A345162 counts normal partitions w/o alt permutation, complement A345163.
%Y A345166 A345170 counts partitions w/ alt permutation, ranked by A345172.
%Y A345166 Cf. A000070, A103919, A335126, A344604, A344607, A344615, A344740, A344742, A345164, A345166, A345168, A345192, A348379.
%K A345166 nonn
%O A345166 0,14
%A A345166 _Gus Wiseman_, Jun 13 2021
%E A345166 a(26) onwards from _Andrew Howroyd_, Jan 15 2024