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A350356 Numbers k such that the k-th composition in standard order is down/up.

%I A350356 #7 Jan 18 2022 05:57:11
%S A350356 0,1,2,4,5,8,9,16,17,18,22,32,33,34,38,44,45,64,65,66,68,70,76,77,88,
%T A350356 89,128,129,130,132,134,140,141,148,152,153,176,177,178,182,256,257,
%U A350356 258,260,262,264,268,269,276,280,281,296,297,304,305,306,310,352,353
%N A350356 Numbers k such that the k-th composition in standard order is down/up.
%C A350356 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 A350356 A composition is down/up if it is alternately strictly increasing and strictly decreasing, starting with a decrease. For example, the partition (3,2,2,2,1) has no down/up permutations, even though it does have the anti-run permutation (2,1,2,3,2).
%H A350356 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%H A350356 Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%F A350356 A345167 = A350355 \/ A350356.
%e A350356 The terms together with the corresponding compositions begin:
%e A350356    0: ()
%e A350356    1: (1)
%e A350356    2: (2)
%e A350356    4: (3)
%e A350356    5: (2,1)
%e A350356    8: (4)
%e A350356    9: (3,1)
%e A350356   16: (5)
%e A350356   17: (4,1)
%e A350356   18: (3,2)
%e A350356   22: (2,1,2)
%e A350356   32: (6)
%e A350356   33: (5,1)
%e A350356   34: (4,2)
%e A350356   38: (3,1,2)
%e A350356   44: (2,1,3)
%e A350356   45: (2,1,2,1)
%t A350356 doupQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<y[[m+1]],y[[m]]>y[[m+1]]],{m,1,Length[y]-1}];
%t A350356 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A350356 Select[Range[0,100],doupQ[stc[#]]&]
%Y A350356 The case of permutations is counted by A000111.
%Y A350356 These compositions are counted by A025049, up/down A025048.
%Y A350356 The strict case is counted by A129838, undirected A349054.
%Y A350356 The weak version is counted by A129853, up/down A129852.
%Y A350356 The version for anti-runs is A333489, a superset, complement A348612.
%Y A350356 This is the down/up case of A345167, counted by A025047.
%Y A350356 Counting patterns of this type gives A350354.
%Y A350356 The up/down version is A350355.
%Y A350356 A001250 counts alternating permutations, complement A348615.
%Y A350356 A003242 counts anti-run compositions.
%Y A350356 A011782 counts compositions, unordered A000041.
%Y A350356 A345192 counts non-alternating compositions, ranked by A345168.
%Y A350356 A349052 counts weakly alternating compositions, complement A349053.
%Y A350356 A349057 ranks non-weakly alternating compositions.
%Y A350356 Statistics of standard compositions:
%Y A350356 - Length is A000120.
%Y A350356 - Sum is A070939.
%Y A350356 - Heinz number is A333219.
%Y A350356 - Number of maximal anti-runs is A333381.
%Y A350356 - Number of distinct parts is A334028.
%Y A350356 Classes of standard compositions:
%Y A350356 - Partitions are A114994, strict A333256.
%Y A350356 - Multisets are A225620, strict A333255.
%Y A350356 - Strict compositions are A233564.
%Y A350356 - Constant compositions are A272919.
%Y A350356 - Patterns are A333217.
%Y A350356 Cf. A008965, A049774, A095684, A106356, A238279, A344604, A344614, A344615, A345169, A345170, A345172, A349051, A349799.
%K A350356 nonn
%O A350356 1,3
%A A350356 _Gus Wiseman_, Jan 15 2022