A373953 - OEIS

%I #5 Jun 27 2024 22:25:59

%S 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 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 7,7,7,5,7,4,5,6,7,7,6,4,7,7,7,6,5,3,5

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

%C 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 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 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 a(n) = A029837(A373948(n)).

%e The standard compositions and their compressions and compression sums begin:

%e    0: ()        --> ()      --> 0

%e    1: (1)       --> (1)     --> 1

%e    2: (2)       --> (2)     --> 2

%e    3: (1,1)     --> (1)     --> 1

%e    4: (3)       --> (3)     --> 3

%e    5: (2,1)     --> (2,1)   --> 3

%e    6: (1,2)     --> (1,2)   --> 3

%e    7: (1,1,1)   --> (1)     --> 1

%e    8: (4)       --> (4)     --> 4

%e    9: (3,1)     --> (3,1)   --> 4

%e   10: (2,2)     --> (2)     --> 2

%e   11: (2,1,1)   --> (2,1)   --> 3

%e   12: (1,3)     --> (1,3)   --> 4

%e   13: (1,2,1)   --> (1,2,1) --> 4

%e   14: (1,1,2)   --> (1,2)   --> 3

%e   15: (1,1,1,1) --> (1)     --> 1

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

%t Table[Total[First/@Split[stc[n]]],{n,0,100}]

%Y Positions of 1's are A000225.

%Y Counting partitions by this statistic gives A116861, by length A116608.

%Y For length instead of sum we have A124767, counted by A238279 and A333755.

%Y Compositions counted by this statistic are A373949, opposite A373951.

%Y A037201 gives compression of first differences of primes, halved A373947.

%Y A066099 lists the parts of all compositions in standard order.

%Y A114901 counts compositions with no isolated parts.

%Y A240085 counts compositions with no unique parts.

%Y A333489 ranks anti-runs, counted by A003242.

%Y Cf. A106356, A124762, A238130, A238343, A272919, A285981, A333381, A333382, A333627, A373952, A373954.

%K nonn

%O 0,3

%A _Gus Wiseman_, Jun 25 2024