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A333381 Number of maximal anti-runs of the n-th composition in standard order.

%I A333381 #9 Mar 24 2020 22:30:02
%S A333381 0,1,1,2,1,1,1,3,1,1,2,2,1,1,2,4,1,1,1,2,1,2,1,3,1,1,2,2,2,2,3,5,1,1,
%T A333381 1,2,2,1,1,3,1,1,3,3,1,1,2,4,1,1,1,2,1,2,1,3,2,2,3,3,3,3,4,6,1,1,1,2,
%U A333381 1,1,1,3,1,2,2,2,1,1,2,4,1,1,1,2,2,3,2
%N A333381 Number of maximal anti-runs of the n-th composition in standard order.
%C A333381 Anti-runs are sequences without any adjacent equal terms.
%C A333381 A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.
%C A333381 For n > 0, also one plus the number of adjacent equal pairs in the n-th composition in standard order.
%H A333381 Wikipedia, <a href="https://en.wikipedia.org/wiki/Longest_increasing_subsequence">Longest increasing subsequence</a>
%F A333381 For n > 0, a(n) = A124762(n) + 1.
%e A333381 The 46th composition in standard order is (2,1,1,2), with maximal anti-runs ((2,1),(1,2)), so a(46) = 2.
%t A333381 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333381 Table[Length[Split[stc[n],UnsameQ]],{n,0,100}]
%Y A333381 Anti-runs summing to n are counted by A003242(n).
%Y A333381 A triangle counting maximal anti-runs of compositions is A106356.
%Y A333381 A triangle counting maximal runs of compositions is A238279.
%Y A333381 Partitions whose first differences are an anti-run are A238424.
%Y A333381 All of the following pertain to compositions in standard order (A066099):
%Y A333381 - Adjacent equal pairs are counted by A124762.
%Y A333381 - Weakly decreasing runs are counted by A124765.
%Y A333381 - Weakly increasing runs are counted by A124766.
%Y A333381 - Equal runs are counted by A124767.
%Y A333381 - Strictly increasing runs are counted by A124768.
%Y A333381 - Strictly decreasing runs are counted by A124769.
%Y A333381 - Strict compositions are ranked by A233564.
%Y A333381 - Constant compositions are ranked by A272919.
%Y A333381 - Normal compositions are ranked by A333217.
%Y A333381 - Adjacent unequal pairs are counted by A333382.
%Y A333381 - Anti-runs are ranked by A333489.
%Y A333381 Cf. A000120, A029931, A048793, A059893, A070939, A114994, A225620, A228351.
%K A333381 nonn
%O A333381 0,4
%A A333381 _Gus Wiseman_, Mar 24 2020