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Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.
6

%I #4 Mar 21 2020 16:36:01

%S 0,1,3,5,7,11,15,21,23,31,37,43,47,63,75,85,87,95,127,149,151,171,175,

%T 191,255,293,299,303,341,343,351,383,511,549,587,597,599,607,683,687,

%U 703,767,1023,1099,1173,1175,1195,1199,1215,1365,1367,1375,1407,1535

%N Numbers k such that the k-th composition in standard order is weakly decreasing and covers an initial interval of positive integers.

%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.

%F Intersection of A333217 and A114994.

%e The sequence of terms together with the corresponding compositions begins:

%e 0: () 127: (1,1,1,1,1,1,1)

%e 1: (1) 149: (3,2,2,1)

%e 3: (1,1) 151: (3,2,1,1,1)

%e 5: (2,1) 171: (2,2,2,1,1)

%e 7: (1,1,1) 175: (2,2,1,1,1,1)

%e 11: (2,1,1) 191: (2,1,1,1,1,1,1)

%e 15: (1,1,1,1) 255: (1,1,1,1,1,1,1,1)

%e 21: (2,2,1) 293: (3,3,2,1)

%e 23: (2,1,1,1) 299: (3,2,2,1,1)

%e 31: (1,1,1,1,1) 303: (3,2,1,1,1,1)

%e 37: (3,2,1) 341: (2,2,2,2,1)

%e 43: (2,2,1,1) 343: (2,2,2,1,1,1)

%e 47: (2,1,1,1,1) 351: (2,2,1,1,1,1,1)

%e 63: (1,1,1,1,1,1) 383: (2,1,1,1,1,1,1,1)

%e 75: (3,2,1,1) 511: (1,1,1,1,1,1,1,1,1)

%e 85: (2,2,2,1) 549: (4,3,2,1)

%e 87: (2,2,1,1,1) 587: (3,3,2,1,1)

%e 95: (2,1,1,1,1,1) 597: (3,2,2,2,1)

%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];

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

%t Select[Range[0,1000],normQ[stc[#]]&&GreaterEqual@@stc[#]&]

%Y Sequences covering an initial interval are counted by A000670.

%Y Compositions in standard order are A066099.

%Y Weakly decreasing runs are counted by A124765.

%Y Removing the covering condition gives A114994.

%Y Removing the ordering condition gives A333217.

%Y The strictly decreasing case is A246534.

%Y The unequal version is A333218.

%Y The weakly increasing version is A333379.

%Y Cf. A000120, A000225, A029931, A048793, A070939, A164894, A225620, A228351, A233564, A272919, A333219.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 21 2020