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A350355 Numbers k such that the k-th composition in standard order is up/down.

%I A350355 #8 Jan 18 2022 05:56:58
%S A350355 0,1,2,4,6,8,12,13,16,20,24,25,32,40,41,48,49,50,54,64,72,80,81,82,96,
%T A350355 97,98,102,108,109,128,144,145,160,161,162,166,192,193,194,196,198,
%U A350355 204,205,216,217,256,272,288,289,290,320,321,322,324,326,332,333,384
%N A350355 Numbers k such that the k-th composition in standard order is up/down.
%C A350355 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%C A350355 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).
%F A350355 A345167 = A350355 \/ A350356.
%e A350355 The terms together with the corresponding compositions begin:
%e A350355    0: ()
%e A350355    1: (1)
%e A350355    2: (2)
%e A350355    4: (3)
%e A350355    6: (1,2)
%e A350355    8: (4)
%e A350355   12: (1,3)
%e A350355   13: (1,2,1)
%e A350355   16: (5)
%e A350355   20: (2,3)
%e A350355   24: (1,4)
%e A350355   25: (1,3,1)
%e A350355   32: (6)
%e A350355   40: (2,4)
%e A350355   41: (2,3,1)
%e A350355   48: (1,5)
%e A350355   49: (1,4,1)
%e A350355   50: (1,3,2)
%e A350355   54: (1,2,1,2)
%t A350355 updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]<y[[m+1]]],{m,1,Length[y]-1}];
%t A350355 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A350355 Select[Range[0,100],updoQ[stc[#]]&]
%Y A350355 The case of permutations is counted by A000111.
%Y A350355 These compositions are counted by A025048, down/up A025049.
%Y A350355 The strict case is counted by A129838, undirected A349054.
%Y A350355 The weak version is counted by A129852, down/up A129853.
%Y A350355 The version for anti-runs is A333489, a superset, complement A348612.
%Y A350355 This is the up/down case of A345167, counted by A025047.
%Y A350355 Counting patterns of this type gives A350354.
%Y A350355 The down/up version is A350356.
%Y A350355 A001250 counts alternating permutations, complement A348615.
%Y A350355 A003242 counts anti-run compositions.
%Y A350355 A011782 counts compositions, unordered A000041.
%Y A350355 A345192 counts non-alternating compositions, ranked by A345168.
%Y A350355 A349052 counts weakly alternating compositions, complement A349053.
%Y A350355 A349057 ranks non-weakly alternating compositions.
%Y A350355 Statistics of standard compositions:
%Y A350355 - Length is A000120.
%Y A350355 - Sum is A070939.
%Y A350355 - Heinz number is A333219.
%Y A350355 - Number of maximal anti-runs is A333381.
%Y A350355 - Number of distinct parts is A334028.
%Y A350355 Classes of standard compositions:
%Y A350355 - Partitions are A114994, strict A333256.
%Y A350355 - Multisets are A225620, strict A333255.
%Y A350355 - Strict compositions are A233564.
%Y A350355 - Constant compositions are A272919.
%Y A350355 - Patterns are A333217.
%Y A350355 Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.
%K A350355 nonn
%O A350355 1,3
%A A350355 _Gus Wiseman_, Jan 15 2022