%I #27 Sep 06 2023 01:39:39
%S 1,1,1,2,3,5,6,10,14,19,25,36,48,64,84,111,146,191,244,315,404,515,
%T 651,823,1035,1295,1616,2011,2492,3076,3787,4650,5695,6952,8463,10280,
%U 12460,15059,18162,21858,26254,31463,37641,44933,53554,63704,75653,89683,106162,125445,148020
%N Number of integer partitions of n with an alternating permutation.
%C 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).
%C 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).
%H Joseph Likar, <a href="/A345170/b345170.txt">Table of n, a(n) for n = 0..1000</a>
%e The a(1) = 1 through a(8) = 14 partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (21) (31) (32) (42) (43) (53)
%e (211) (41) (51) (52) (62)
%e (221) (321) (61) (71)
%e (311) (411) (322) (332)
%e (2211) (331) (422)
%e (421) (431)
%e (511) (521)
%e (3211) (611)
%e (22111) (3221)
%e (3311)
%e (4211)
%e (22211)
%e (32111)
%t wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],wigQ]!={}&]],{n,0,15}]
%Y Includes all strict partitions A000009.
%Y Including twins (x,x) gives A344740.
%Y The normal case is A345163 (complement: A345162).
%Y The complement is counted by A345165, ranked by A345171.
%Y The Heinz numbers of these partitions are A345172.
%Y The version for factorizations is A348379.
%Y A000041 counts integer partitions.
%Y A001250 counts alternating permutations.
%Y A003242 counts anti-run compositions.
%Y A005649 counts anti-run patterns.
%Y A025047 counts alternating compositions (ascend: A025048, descend: A025049).
%Y A325534 counts separable partitions, ranked by A335433.
%Y A325535 counts inseparable partitions, ranked by A335448.
%Y A344604 counts alternating compositions with twins.
%Y Cf. A000070, A103919, A335126, A344605, A344653, A344654, A344742, A345164, A345166, A345167, A345168, A345195.
%K nonn
%O 0,4
%A _Gus Wiseman_, Jun 13 2021
%E a(26)-a(32) from _Robert Price_, Jun 23 2021
%E a(33)-a(48) from _Alois P. Heinz_, Jun 23 2021
%E a(49) onwards from _Joseph Likar_, Sep 05 2023