%I #6 Jun 28 2024 10:30:56

%S 1,1,0,1,1,0,3,0,1,0,4,2,1,1,0,7,4,4,0,1,0,14,5,6,5,1,1,0,23,14,10,10,

%T 6,0,1,0,39,26,29,12,14,6,1,1,0,71,46,54,40,19,16,9,0,1,0,124,92,96,

%U 82,64,22,22,8,1,1,0,214,176,204,144,137,82,30,26,10,0,1,0

%N Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of n - k.

%e Triangle begins:

%e 1

%e 1 0

%e 1 1 0

%e 3 0 1 0

%e 4 2 1 1 0

%e 7 4 4 0 1 0

%e 14 5 6 5 1 1 0

%e 23 14 10 10 6 0 1 0

%e 39 26 29 12 14 6 1 1 0

%e 71 46 54 40 19 16 9 0 1 0

%e 124 92 96 82 64 22 22 8 1 1 0

%e Row n = 6 counts the following compositions:

%e (6) (411) (3111) (33) (222) (111111) .

%e (51) (114) (1113) (2211)

%e (15) (1311) (1221) (1122)

%e (42) (1131) (12111) (21111)

%e (24) (2112) (11211) (11112)

%e (141) (11121)

%e (321)

%e (312)

%e (231)

%e (213)

%e (132)

%e (123)

%e (2121)

%e (1212)

%e For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,2).

%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#]]==n-k&]],{n,0,10},{k,0,n}]

%Y Column k = 0 is A003242 (anti-runs or compressed compositions).

%Y Row-sums are A011782.

%Y Same as A373949 with rows reversed.

%Y Column k = 1 is A373950.

%Y This statistic is represented by A373954, difference A373953.

%Y A114901 counts compositions with no isolated parts.

%Y A116861 counts partitions by compressed sum, by compressed length A116608.

%Y A124767 counts runs in standard compositions, anti-runs A333381.

%Y A240085 counts compositions with no unique parts.

%Y A333755 counts compositions by compressed length.

%Y A373948 represents the run-compression transformation.

%Y Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333382, A333489, A373952.

%K nonn,tabl

%O 0,7

%A _Gus Wiseman_, Jun 28 2024