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A025048 Number of up/down (initially ascending) compositions of n.

%I A025048 #28 Apr 29 2025 14:33:32
%S A025048 1,1,1,2,3,4,7,11,16,26,41,64,100,158,247,389,612,960,1509,2372,3727,
%T A025048 5858,9207,14468,22738,35737,56164,88268,138726,218024,342652,538524,
%U A025048 846358,1330160,2090522,3285526,5163632,8115323,12754288,20045027,31503382
%N A025048 Number of up/down (initially ascending) compositions of n.
%C A025048 Original name was: Ascending wiggly sums: number of sums adding to n in which terms alternately increase and decrease.
%C A025048 A composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2). - _Gus Wiseman_, Jan 15 2022
%H A025048 Alois P. Heinz, <a href="/A025048/b025048.txt">Table of n, a(n) for n = 0..1000</a>
%H A025048 Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%F A025048 a(n) = 1 + A025047(n) - A025049(n) = Sum_k A059882(n,k). - _Henry Bottomley_, Feb 05 2001
%F A025048 a(n) ~ c * d^n, where d = 1.571630806607064114100138865739690782401305155950789062725011227781640624..., c = 0.4408955566119650057730070154620695491718230084159159991449729825619... . - _Vaclav Kotesovec_, Sep 12 2014
%e A025048 From _Gus Wiseman_, Jan 15 2022: (Start)
%e A025048 The a(1) = 1 through a(7) = 11 up/down compositions:
%e A025048   (1)  (2)  (3)    (4)      (5)      (6)        (7)
%e A025048             (1,2)  (1,3)    (1,4)    (1,5)      (1,6)
%e A025048                    (1,2,1)  (2,3)    (2,4)      (2,5)
%e A025048                             (1,3,1)  (1,3,2)    (3,4)
%e A025048                                      (1,4,1)    (1,4,2)
%e A025048                                      (2,3,1)    (1,5,1)
%e A025048                                      (1,2,1,2)  (2,3,2)
%e A025048                                                 (2,4,1)
%e A025048                                                 (1,2,1,3)
%e A025048                                                 (1,3,1,2)
%e A025048                                                 (1,2,1,2,1)
%e A025048 (End)
%t A025048 updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]<y[[m+1]]],{m,1,Length[y]-1}];
%t A025048 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],updoQ]],{n,0,15}] (* _Gus Wiseman_, Jan 15 2022 *)
%Y A025048 The case of permutations is A000111.
%Y A025048 The undirected version is A025047, ranked by A345167.
%Y A025048 The down/up version is A025049, ranked by A350356.
%Y A025048 The strict case is A129838, undirected A349054.
%Y A025048 The weak version is A129852, down/up A129853.
%Y A025048 The version for patterns is A350354.
%Y A025048 These compositions are ranked by A350355.
%Y A025048 A001250 counts alternating permutations, complement A348615.
%Y A025048 A003242 counts Carlitz compositions, complement A261983.
%Y A025048 A011782 counts compositions, unordered A000041.
%Y A025048 A325534 counts separable partitions, complement A325535.
%Y A025048 A345192 counts non-alternating compositions, ranked by A345168.
%Y A025048 A345194 counts alternating patterns, complement A350252.
%Y A025048 A349052 counts weakly alternating compositions, complement A349053.
%Y A025048 Cf. A008965, A049774, A128761, A344604, A344605, A344614, A344615, A345195, A349057, A349800.
%K A025048 nonn
%O A025048 0,4
%A A025048 _David W. Wilson_
%E A025048 Name and offset changed by _Gus Wiseman_, Jan 15 2022