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A349796
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Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.
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12
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0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 8, 15, 23, 37, 52, 80, 109, 156, 208, 289, 378, 509, 654, 865, 1098, 1425, 1789, 2290, 2852, 3603, 4450, 5569, 6830, 8467, 10321, 12701, 15393, 18805, 22678, 27535, 33057, 39908, 47701, 57304, 68226, 81572, 96766, 115212, 136201
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OFFSET
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0,9
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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The a(7) = 1 through a(11) = 15 partitions:
(3211) (4211) (3321) (5311) (4322)
(32111) (4311) (6211) (4421)
(5211) (32221) (5411)
(42111) (33211) (6311)
(321111) (43111) (7211)
(52111) (42221)
(421111) (43211)
(3211111) (53111)
(62111)
(322211)
(332111)
(431111)
(521111)
(4211111)
(32111111)
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MATHEMATICA
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whkQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]<=y[[m+1]], y[[m]]>=y[[m+1]]], {m, 1, Length[y]-1}];
Table[Length[Select[IntegerPartitions[n], !whkQ[#]&&!whkQ[-#]&&!UnsameQ@@#&]], {n, 0, 30}]
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CROSSREFS
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Counting all non-strict partitions gives A047967.
The strict instead of non-strict version is A347548, ranked by A350352.
The version for compositions allowing strict is A349053, ranked by A349057.
The complement in non-strict partitions is A349795.
A003242 = Carlitz (anti-run) compositions.
A096441 = weakly alternating 0-appended partitions.
A345170 = partitions w/ an alternating permutation, ranked by A345172.
A349052 = weakly alternating compositions.
A349056 = weakly alternating permutations of prime indices.
A349798 = weakly but not strongly alternating permutations of prime indices.
Cf. A000111, A002865, A117298, A117989, A129852, A129853, A345165, A345192, A349054, A349059, A349801.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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