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A373953 Sum of run-compression of the n-th integer composition in standard order.

%I A373953 #5 Jun 27 2024 22:25:59
%S A373953 0,1,2,1,3,3,3,1,4,4,2,3,4,4,3,1,5,5,5,4,5,3,5,3,5,5,3,4,4,4,3,1,6,6,
%T A373953 6,5,3,6,6,4,6,6,2,3,6,6,5,3,6,6,6,5,6,4,6,4,5,5,3,4,4,4,3,1,7,7,7,6,
%U A373953 7,7,7,5,7,4,5,6,7,7,6,4,7,7,7,6,5,3,5
%N A373953 Sum of run-compression of the n-th integer composition in standard order.
%C A373953 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 A373953 We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).
%H A373953 Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vTCPiJVFUXN8IqfLlCXkgP15yrGWeRhFS4ozST5oA4Bl2PYS-XTA3sGsAEXvwW-B0ealpD8qnoxFqN3/pub">Statistics, classes, and transformations of standard compositions</a>
%F A373953 a(n) = A029837(A373948(n)).
%e A373953 The standard compositions and their compressions and compression sums begin:
%e A373953    0: ()        --> ()      --> 0
%e A373953    1: (1)       --> (1)     --> 1
%e A373953    2: (2)       --> (2)     --> 2
%e A373953    3: (1,1)     --> (1)     --> 1
%e A373953    4: (3)       --> (3)     --> 3
%e A373953    5: (2,1)     --> (2,1)   --> 3
%e A373953    6: (1,2)     --> (1,2)   --> 3
%e A373953    7: (1,1,1)   --> (1)     --> 1
%e A373953    8: (4)       --> (4)     --> 4
%e A373953    9: (3,1)     --> (3,1)   --> 4
%e A373953   10: (2,2)     --> (2)     --> 2
%e A373953   11: (2,1,1)   --> (2,1)   --> 3
%e A373953   12: (1,3)     --> (1,3)   --> 4
%e A373953   13: (1,2,1)   --> (1,2,1) --> 4
%e A373953   14: (1,1,2)   --> (1,2)   --> 3
%e A373953   15: (1,1,1,1) --> (1)     --> 1
%t A373953 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A373953 Table[Total[First/@Split[stc[n]]],{n,0,100}]
%Y A373953 Positions of 1's are A000225.
%Y A373953 Counting partitions by this statistic gives A116861, by length A116608.
%Y A373953 For length instead of sum we have A124767, counted by A238279 and A333755.
%Y A373953 Compositions counted by this statistic are A373949, opposite A373951.
%Y A373953 A037201 gives compression of first differences of primes, halved A373947.
%Y A373953 A066099 lists the parts of all compositions in standard order.
%Y A373953 A114901 counts compositions with no isolated parts.
%Y A373953 A240085 counts compositions with no unique parts.
%Y A373953 A333489 ranks anti-runs, counted by A003242.
%Y A373953 Cf. A106356, A124762, A238130, A238343, A272919, A285981, A333381, A333382, A333627, A373952, A373954.
%K A373953 nonn
%O A373953 0,3
%A A373953 _Gus Wiseman_, Jun 25 2024