cp's OEIS Frontend

A374251 Irregular triangle read by rows where row n is the run-compression of the n-th composition in standard order.

%I A374251 #6 Jul 11 2024 08:38:06
%S A374251 1,2,1,3,2,1,1,2,1,4,3,1,2,2,1,1,3,1,2,1,1,2,1,5,4,1,3,2,3,1,2,3,2,1,
%T A374251 2,1,2,2,1,1,4,1,3,1,1,2,1,2,1,1,3,1,2,1,1,2,1,6,5,1,4,2,4,1,3,3,2,1,
%U A374251 3,1,2,3,1,2,4,2,3,1,2,2,1,2,1,3
%N A374251 Irregular triangle read by rows where row n is the run-compression of the n-th composition in standard order.
%C A374251 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 A374251 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, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).
%H A374251 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 A374251 The standard compositions and their run-compressions begin:
%e A374251    0: ()        --> ()
%e A374251    1: (1)       --> (1)
%e A374251    2: (2)       --> (2)
%e A374251    3: (1,1)     --> (1)
%e A374251    4: (3)       --> (3)
%e A374251    5: (2,1)     --> (2,1)
%e A374251    6: (1,2)     --> (1,2)
%e A374251    7: (1,1,1)   --> (1)
%e A374251    8: (4)       --> (4)
%e A374251    9: (3,1)     --> (3,1)
%e A374251   10: (2,2)     --> (2)
%e A374251   11: (2,1,1)   --> (2,1)
%e A374251   12: (1,3)     --> (1,3)
%e A374251   13: (1,2,1)   --> (1,2,1)
%e A374251   14: (1,1,2)   --> (1,2)
%e A374251   15: (1,1,1,1) --> (1)
%t A374251 stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A374251 Table[First/@Split[stc[n]],{n,100}]
%Y A374251 Last column is A001511.
%Y A374251 First column is A065120.
%Y A374251 Row-lengths are A124767.
%Y A374251 Using prime indices we get A304038, row-sums A066328.
%Y A374251 Row n has A334028(n) distinct elements.
%Y A374251 Rows are ranked by A373948 (standard order).
%Y A374251 Row-sums are A373953.
%Y A374251 A003242 counts run-compressed compositions, i.e., anti-runs, ranks A333489.
%Y A374251 A007947 (squarefree kernel) represents run-compression of multisets.
%Y A374251 A037201 run-compresses first differences of primes, halved A373947.
%Y A374251 A066099 lists the parts of compositions in standard order.
%Y A374251 A116861 counts partitions by sum of run-compression.
%Y A374251 A238279 and A333755 count compositions by number of runs.
%Y A374251 A373949 counts compositions by sum of run-compression, opposite A373951.
%Y A374251 Cf. A000120, A070939, A106356, A124762, A233564, A238130, A238343, A272919, A333381, A333382, A333627, A373954, A374250.
%K A374251 nonn,tabf
%O A374251 1,2
%A A374251 _Gus Wiseman_, Jul 09 2024