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A373955 Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.

%I A373955 #5 Jun 30 2024 00:34:26
%S A373955 3,11,14,19,27,28,29,35,46,51,56,57,67,75,78,83,91,92,93,99,110,112,
%T A373955 113,114,116,118,131,139,142,155,156,157,163,179,184,185,195,203,206,
%U A373955 211,219,220,221,224,225,226,229,230,232,233,236,237,259,267,270,275
%N A373955 Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.
%C A373955 Also numbers k such that the excess compression of the k-th integer composition in standard order is 1.
%C A373955 The k-th composition in standard order (graded reverse-lexicographic, 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. This gives a bijective correspondence between nonnegative integers and integer compositions.
%C A373955 postn of 1 in
%H A373955 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%e A373955 The terms and corresponding compositions begin:
%e A373955     3: (1,1)
%e A373955    11: (2,1,1)
%e A373955    14: (1,1,2)
%e A373955    19: (3,1,1)
%e A373955    27: (1,2,1,1)
%e A373955    28: (1,1,3)
%e A373955    29: (1,1,2,1)
%e A373955    35: (4,1,1)
%e A373955    46: (2,1,1,2)
%e A373955    51: (1,3,1,1)
%e A373955    56: (1,1,4)
%e A373955    57: (1,1,3,1)
%e A373955    67: (5,1,1)
%e A373955    75: (3,2,1,1)
%e A373955    78: (3,1,1,2)
%e A373955    83: (2,3,1,1)
%e A373955    91: (2,1,2,1,1)
%e A373955    92: (2,1,1,3)
%e A373955    93: (2,1,1,2,1)
%e A373955    99: (1,4,1,1)
%t A373955 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A373955 Select[Range[100],Total[stc[#]] == Total[First/@Split[stc[#]]]+1&]
%Y A373955 These compositions are counted by A373950.
%Y A373955 Positions of ones in A373954.
%Y A373955 A003242 counts compressed compositions (or anti-runs).
%Y A373955 A114901 counts compositions with no isolated parts.
%Y A373955 A116861 counts partitions by compressed sum, by compressed length A116608.
%Y A373955 A124767 counts runs in standard compositions, anti-runs A333381.
%Y A373955 A240085 counts compositions with no unique parts.
%Y A373955 A333755 counts compositions by compressed length.
%Y A373955 A373948 encodes compression using compositions in standard order.
%Y A373955 A373949 counts compositions by compression-sum.
%Y A373955 A373953 gives compression-sum of standard compositions.
%Y A373955 Cf. A106356, A124762, A238130, A238279, A238343, A285981, A333382, A333489, A373951, A373952.
%K A373955 nonn
%O A373955 1,1
%A A373955 _Gus Wiseman_, Jun 29 2024