A353858 - OEIS

%I #6 Jun 17 2022 22:12:40

%S 0,1,2,2,5,2,8,2,20,5,8,2,78,2,8,8,223,2,179,2,142,8,8,2,4808

%N Number of integer compositions of n with run-sum trajectory ending in a singleton.

%C Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sums (cf. A353847) until an anti-run composition (A003242) is reached. For example, the composition (2,2,1,1,2) is counted under a(8) because it has the following run-sum trajectory: (2,2,1,1,2) -> (4,2,2) -> (4,4) -> (8).

%e The a(0) = 0 through a(8) = 20 compositions:

%e   .  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)

%e           (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)

%e                        (112)            (222)                (224)

%e                        (211)            (1113)               (422)

%e                        (1111)           (2112)               (1124)

%e                                         (3111)               (2114)

%e                                         (11211)              (2222)

%e                                         (111111)             (4112)

%e                                                              (4211)

%e                                                              (11114)

%e                                                              (21122)

%e                                                              (22112)

%e                                                              (41111)

%e                                                              (111122)

%e                                                              (112112)

%e                                                              (211211)

%e                                                              (221111)

%e                                                              (1111211)

%e                                                              (1121111)

%e                                                              (11111111)

%t Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], Length[FixedPoint[Total/@Split[#]&,#]]==1&]],{n,0,15}]

%Y The version for partitions is A353845, ranked by A353844.

%Y The trajectory itself is A353853, last part A353855.

%Y The lengths of trajectories of standard compositions are A353854.

%Y This is column k = 1 of A353856, for partitions A353843.

%Y These compositions are ranked by A353857.

%Y A011782 counts compositions.

%Y A066099 lists compositions in standard order.

%Y A238279 and A333755 count compositions by number of runs.

%Y A275870 counts collapsible partitions, ranked by A300273.

%Y A333489 ranks anti-runs, counted by A003242 (complement A261983).

%Y A353840-A353846 pertain to partition run-sum trajectory.

%Y A353847 represents the run-sums of a composition, partitions A353832.

%Y A353851 counts compositions with equal run-sums, ranked by A353848.

%Y A353859 counts compositions by length of run-sum trajectory.

%Y A353860 counts collapsible compositions.

%Y A353932 lists run-sums of standard compositions.

%Y Cf. A072639, A237685, A304442, A304465, A318928, A353833, A353841, A353849, A353850, A353852.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jun 17 2022