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Numbers k such that the k-th composition in standard order is weakly increasing and covers an initial interval of positive integers.
8

%I #6 Mar 21 2020 16:35:52

%S 0,1,3,6,7,14,15,26,30,31,52,58,62,63,106,116,122,126,127,212,234,244,

%T 250,254,255,420,426,468,490,500,506,510,511,840,852,932,938,980,1002,

%U 1012,1018,1022,1023,1700,1706,1864,1876,1956,1962,2004,2026,2036,2042

%N Numbers k such that the k-th composition in standard order is weakly increasing 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 A225620.

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

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

%e 1: (1) 212: (1,2,2,3)

%e 3: (1,1) 234: (1,1,2,2,2)

%e 6: (1,2) 244: (1,1,1,2,3)

%e 7: (1,1,1) 250: (1,1,1,1,2,2)

%e 14: (1,1,2) 254: (1,1,1,1,1,1,2)

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

%e 26: (1,2,2) 420: (1,2,3,3)

%e 30: (1,1,1,2) 426: (1,2,2,2,2)

%e 31: (1,1,1,1,1) 468: (1,1,2,2,3)

%e 52: (1,2,3) 490: (1,1,1,2,2,2)

%e 58: (1,1,2,2) 500: (1,1,1,1,2,3)

%e 62: (1,1,1,1,2) 506: (1,1,1,1,1,2,2)

%e 63: (1,1,1,1,1,1) 510: (1,1,1,1,1,1,1,2)

%e 106: (1,2,2,2) 511: (1,1,1,1,1,1,1,1,1)

%e 116: (1,1,2,3) 840: (1,2,3,4)

%e 122: (1,1,1,2,2) 852: (1,2,2,2,3)

%e 126: (1,1,1,1,1,2) 932: (1,1,2,3,3)

%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[#]]&&LessEqual@@stc[#]&]

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

%Y Compositions in standard order are A066099.

%Y Weakly increasing runs are counted by A124766.

%Y Removing the covering condition gives A225620.

%Y Removing the ordering condition gives A333217.

%Y The strictly increasing case is A164894.

%Y The strictly decreasing version is A246534.

%Y The unequal version is A333218.

%Y The weakly decreasing version is A333380.

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

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 21 2020