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 A345170 Number of integer partitions of n with an alternating permutation. 58
 1, 1, 1, 2, 3, 5, 6, 10, 14, 19, 25, 36, 48, 64, 84, 111, 146, 191, 244, 315, 404, 515, 651, 823, 1035, 1295, 1616, 2011, 2492, 3076, 3787, 4650, 5695, 6952, 8463, 10280, 12460, 15059, 18162, 21858, 26254, 31463, 37641, 44933, 53554, 63704, 75653, 89683, 106162, 125445, 148020 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS First differs from A325534 at a(10) = 25, A325534(10) = 26. The first separable partition without an alternating permutation is (3,2,2,2,1). A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,3,2,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,3,2,1,2), (2,3,2,1,2,3,2), and (2,1,2,3,2,3,2). LINKS Joseph Likar, Table of n, a(n) for n = 0..1000 EXAMPLE The a(1) = 1 through a(8) = 14 partitions: (1) (2) (3) (4) (5) (6) (7) (8) (21) (31) (32) (42) (43) (53) (211) (41) (51) (52) (62) (221) (321) (61) (71) (311) (411) (322) (332) (2211) (331) (422) (421) (431) (511) (521) (3211) (611) (22111) (3221) (3311) (4211) (22211) (32111) 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[#], wigQ]!={}&]], {n, 0, 15}] CROSSREFS Includes all strict partitions A000009. Including twins (x,x) gives A344740. The normal case is A345163 (complement: A345162). The complement is counted by A345165, ranked by A345171. The Heinz numbers of these partitions are A345172. The version for factorizations is A348379. A000041 counts integer partitions. A001250 counts alternating permutations. A003242 counts anti-run compositions. A005649 counts anti-run patterns. A025047 counts alternating compositions (ascend: A025048, descend: A025049). A325534 counts separable partitions, ranked by A335433. A325535 counts inseparable partitions, ranked by A335448. A344604 counts alternating compositions with twins. Cf. A000070, A103919, A335126, A344605, A344653, A344654, A344742, A345164, A345166, A345167, A345168, A345195. Sequence in context: A014853 A131627 A325712 * A237822 A325713 A325714 Adjacent sequences: A345167 A345168 A345169 * A345171 A345172 A345173 KEYWORD nonn AUTHOR Gus Wiseman, Jun 13 2021 EXTENSIONS a(26)-a(32) from Robert Price, Jun 23 2021 a(33)-a(48) from Alois P. Heinz, Jun 23 2021 a(49) onwards from Joseph Likar, Sep 05 2023 STATUS approved

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Last modified September 28 02:54 EDT 2023. Contains 365714 sequences. (Running on oeis4.)