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A345170 Number of integer partitions of n with an alternating permutation.

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