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A333489 Numbers k such that the k-th composition in standard order is an anti-run (no adjacent equal parts).

%I A333489 #7 Mar 21 2022 04:21:58
%S A333489 0,1,2,4,5,6,8,9,12,13,16,17,18,20,22,24,25,32,33,34,37,38,40,41,44,
%T A333489 45,48,49,50,52,54,64,65,66,68,69,70,72,76,77,80,81,82,88,89,96,97,98,
%U A333489 101,102,104,105,108,109,128,129,130,132,133,134,137,140,141
%N A333489 Numbers k such that the k-th composition in standard order is an anti-run (no adjacent equal parts).
%C A333489 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.
%H A333489 Vaclav Kotesovec, <a href="/A333489/b333489.txt">Table of n, a(n) for n = 1..10000</a>
%e A333489 The sequence together with the corresponding compositions begins:
%e A333489     0: ()          33: (5,1)         70: (4,1,2)
%e A333489     1: (1)         34: (4,2)         72: (3,4)
%e A333489     2: (2)         37: (3,2,1)       76: (3,1,3)
%e A333489     4: (3)         38: (3,1,2)       77: (3,1,2,1)
%e A333489     5: (2,1)       40: (2,4)         80: (2,5)
%e A333489     6: (1,2)       41: (2,3,1)       81: (2,4,1)
%e A333489     8: (4)         44: (2,1,3)       82: (2,3,2)
%e A333489     9: (3,1)       45: (2,1,2,1)     88: (2,1,4)
%e A333489    12: (1,3)       48: (1,5)         89: (2,1,3,1)
%e A333489    13: (1,2,1)     49: (1,4,1)       96: (1,6)
%e A333489    16: (5)         50: (1,3,2)       97: (1,5,1)
%e A333489    17: (4,1)       52: (1,2,3)       98: (1,4,2)
%e A333489    18: (3,2)       54: (1,2,1,2)    101: (1,3,2,1)
%e A333489    20: (2,3)       64: (7)          102: (1,3,1,2)
%e A333489    22: (2,1,2)     65: (6,1)        104: (1,2,4)
%e A333489    24: (1,4)       66: (5,2)        105: (1,2,3,1)
%e A333489    25: (1,3,1)     68: (4,3)        108: (1,2,1,3)
%e A333489    32: (6)         69: (4,2,1)      109: (1,2,1,2,1)
%t A333489 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333489 Select[Range[0,100],!MatchQ[stc[#],{___,x_,x_,___}]&]
%Y A333489 Anti-runs summing to n are counted by A003242(n).
%Y A333489 A triangle counting maximal anti-runs of compositions is A106356.
%Y A333489 A triangle counting maximal runs of compositions is A238279 or A238130.
%Y A333489 Partitions whose first differences are an anti-run are A238424.
%Y A333489 All of the following pertain to compositions in standard order (A066099):
%Y A333489 - Adjacent equal pairs are counted by A124762.
%Y A333489 - Weakly decreasing runs are counted by A124765.
%Y A333489 - Weakly increasing runs are counted by A124766.
%Y A333489 - Equal runs are counted by A124767.
%Y A333489 - Strictly increasing runs are counted by A124768.
%Y A333489 - Strictly decreasing runs are counted by A124769.
%Y A333489 - Strict compositions are ranked by A233564.
%Y A333489 - Constant compositions are ranked by A272919.
%Y A333489 - Normal compositions are ranked by A333217.
%Y A333489 - Anti-runs are counted by A333381.
%Y A333489 - Adjacent unequal pairs are counted by A333382.
%Y A333489 Cf. A000120, A029931, A048793, A066099, A070939, A228351.
%K A333489 nonn
%O A333489 1,3
%A A333489 _Gus Wiseman_, Mar 28 2020