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Numbers k such that the k-th composition in standard order is an anti-run (no adjacent equal parts).
233

%I #7 Mar 21 2022 04:21:58

%S 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 45,48,49,50,52,54,64,65,66,68,69,70,72,76,77,80,81,82,88,89,96,97,98,

%U 101,102,104,105,108,109,128,129,130,132,133,134,137,140,141

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

%C 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 Vaclav Kotesovec, <a href="/A333489/b333489.txt">Table of n, a(n) for n = 1..10000</a>

%e The sequence together with the corresponding compositions begins:

%e 0: () 33: (5,1) 70: (4,1,2)

%e 1: (1) 34: (4,2) 72: (3,4)

%e 2: (2) 37: (3,2,1) 76: (3,1,3)

%e 4: (3) 38: (3,1,2) 77: (3,1,2,1)

%e 5: (2,1) 40: (2,4) 80: (2,5)

%e 6: (1,2) 41: (2,3,1) 81: (2,4,1)

%e 8: (4) 44: (2,1,3) 82: (2,3,2)

%e 9: (3,1) 45: (2,1,2,1) 88: (2,1,4)

%e 12: (1,3) 48: (1,5) 89: (2,1,3,1)

%e 13: (1,2,1) 49: (1,4,1) 96: (1,6)

%e 16: (5) 50: (1,3,2) 97: (1,5,1)

%e 17: (4,1) 52: (1,2,3) 98: (1,4,2)

%e 18: (3,2) 54: (1,2,1,2) 101: (1,3,2,1)

%e 20: (2,3) 64: (7) 102: (1,3,1,2)

%e 22: (2,1,2) 65: (6,1) 104: (1,2,4)

%e 24: (1,4) 66: (5,2) 105: (1,2,3,1)

%e 25: (1,3,1) 68: (4,3) 108: (1,2,1,3)

%e 32: (6) 69: (4,2,1) 109: (1,2,1,2,1)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[0,100],!MatchQ[stc[#],{___,x_,x_,___}]&]

%Y Anti-runs summing to n are counted by A003242(n).

%Y A triangle counting maximal anti-runs of compositions is A106356.

%Y A triangle counting maximal runs of compositions is A238279 or A238130.

%Y Partitions whose first differences are an anti-run are A238424.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - Adjacent equal pairs are counted by A124762.

%Y - Weakly decreasing runs are counted by A124765.

%Y - Weakly increasing runs are counted by A124766.

%Y - Equal runs are counted by A124767.

%Y - Strictly increasing runs are counted by A124768.

%Y - Strictly decreasing runs are counted by A124769.

%Y - Strict compositions are ranked by A233564.

%Y - Constant compositions are ranked by A272919.

%Y - Normal compositions are ranked by A333217.

%Y - Anti-runs are counted by A333381.

%Y - Adjacent unequal pairs are counted by A333382.

%Y Cf. A000120, A029931, A048793, A066099, A070939, A228351.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 28 2020