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A373948 Run-compression encoded as a transformation of compositions in standard order.

%I A373948 #7 Jun 27 2024 18:13:02
%S A373948 0,1,2,1,4,5,6,1,8,9,2,5,12,13,6,1,16,17,18,9,20,5,22,5,24,25,6,13,12,
%T A373948 13,6,1,32,33,34,17,4,37,38,9,40,41,2,5,44,45,22,5,48,49,50,25,52,13,
%U A373948 54,13,24,25,6,13,12,13,6,1,64,65,66,33,68,69,70,17,72
%N A373948 Run-compression encoded as a transformation of compositions in standard order.
%C A373948 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 A373948 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).
%C A373948 For the present sequence, the a(n)-th composition in standard order is obtained by compressing the n-th composition in standard order.
%H A373948 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 A373948 A029837(a(n)) = A373953(n).
%F A373948 A000120(a(n)) = A124767(n).
%e A373948 The standard compositions and their compressions begin:
%e A373948    0: ()        -->  0: ()
%e A373948    1: (1)       -->  1: (1)
%e A373948    2: (2)       -->  2: (2)
%e A373948    3: (1,1)     -->  1: (1)
%e A373948    4: (3)       -->  4: (3)
%e A373948    5: (2,1)     -->  5: (2,1)
%e A373948    6: (1,2)     -->  6: (1,2)
%e A373948    7: (1,1,1)   -->  1: (1)
%e A373948    8: (4)       -->  8: (4)
%e A373948    9: (3,1)     -->  9: (3,1)
%e A373948   10: (2,2)     -->  2: (2)
%e A373948   11: (2,1,1)   -->  5: (2,1)
%e A373948   12: (1,3)     --> 12: (1,3)
%e A373948   13: (1,2,1)   --> 13: (1,2,1)
%e A373948   14: (1,1,2)   -->  6: (1,2)
%e A373948   15: (1,1,1,1) -->  1: (1)
%t A373948 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A373948 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t A373948 Table[stcinv[First/@Split[stc[n]]],{n,0,30}]
%Y A373948 Positions of 1's are A000225.
%Y A373948 The image is A333489, counted by A003242.
%Y A373948 Sum of standard composition for a(n) is given by A373953, length A124767.
%Y A373948 A037201 gives compression of first differences of primes, halved A373947.
%Y A373948 A066099 lists the parts of all compositions in standard order.
%Y A373948 A114901 counts compositions with no isolated parts.
%Y A373948 A116861 counts partitions by compressed sum, by length A116608.
%Y A373948 A240085 counts compositions with no unique parts.
%Y A373948 A333755 counts compositions by compressed length.
%Y A373948 A373949 counts compositions by compressed sum, opposite A373951.
%Y A373948 Cf. A106356, A124762, A238130, A238279, A272919, A333381, A333382, A333627, A373952, A373954.
%K A373948 nonn
%O A373948 0,3
%A A373948 _Gus Wiseman_, Jun 24 2024