cp's OEIS Frontend

A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.

%I A349796 #6 Jan 07 2022 15:54:26
%S A349796 0,0,0,0,0,0,0,1,2,5,8,15,23,37,52,80,109,156,208,289,378,509,654,865,
%T A349796 1098,1425,1789,2290,2852,3603,4450,5569,6830,8467,10321,12701,15393,
%U A349796 18805,22678,27535,33057,39908,47701,57304,68226,81572,96766,115212,136201
%N A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.
%C A349796 Also the number of non-weakly alternating non-strict integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence involves the somewhat degenerate case where no strict increases are allowed.
%F A349796 a(n) = A349061(n) - A347548(n).
%e A349796 The a(7) = 1 through a(11) = 15 partitions:
%e A349796   (3211)  (4211)   (3321)    (5311)     (4322)
%e A349796           (32111)  (4311)    (6211)     (4421)
%e A349796                    (5211)    (32221)    (5411)
%e A349796                    (42111)   (33211)    (6311)
%e A349796                    (321111)  (43111)    (7211)
%e A349796                              (52111)    (42221)
%e A349796                              (421111)   (43211)
%e A349796                              (3211111)  (53111)
%e A349796                                         (62111)
%e A349796                                         (322211)
%e A349796                                         (332111)
%e A349796                                         (431111)
%e A349796                                         (521111)
%e A349796                                         (4211111)
%e A349796                                         (32111111)
%t A349796 whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
%t A349796 Table[Length[Select[IntegerPartitions[n],!whkQ[#]&&!whkQ[-#]&&!UnsameQ@@#&]],{n,0,30}]
%Y A349796 Counting all non-strict partitions gives A047967.
%Y A349796 Signatures of this type are counted by A274230, complement A027383.
%Y A349796 The strict instead of non-strict version is A347548, ranked by A350352.
%Y A349796 The version for compositions allowing strict is A349053, ranked by A349057.
%Y A349796 Allowing strict partitions gives A349061, complement A349060.
%Y A349796 The complement in non-strict partitions is A349795.
%Y A349796 These partitions are ranked by A350140 = A349794 \ A005117.
%Y A349796 A000041 = integer partitions, strict A000009.
%Y A349796 A001250 = alternating permutations, complement A348615.
%Y A349796 A003242 = Carlitz (anti-run) compositions.
%Y A349796 A025047 = alternating compositions, ranked by A345167.
%Y A349796 A025048/A025049 = directed alternating compositions.
%Y A349796 A096441 = weakly alternating 0-appended partitions.
%Y A349796 A345170 = partitions w/ an alternating permutation, ranked by A345172.
%Y A349796 A349052 = weakly alternating compositions.
%Y A349796 A349056 = weakly alternating permutations of prime indices.
%Y A349796 A349798 = weakly but not strongly alternating permutations of prime indices.
%Y A349796 Cf. A000111, A002865, A117298, A117989, A129852, A129853, A345165, A345192, A349054, A349059, A349801.
%K A349796 nonn
%O A349796 0,9
%A A349796 _Gus Wiseman_, Dec 25 2021