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%I #8 Jun 28 2024 10:31:01
%S 1,0,1,0,1,1,0,1,0,3,0,1,1,2,4,0,1,0,4,4,7,0,1,1,5,6,5,14,0,1,0,6,10,
%T 10,14,23,0,1,1,6,14,12,29,26,39,0,1,0,9,16,19,40,54,46,71,0,1,1,8,22,
%U 22,64,82,96,92,124,0,1,0,10,26,30,82,137,144,204,176,214
%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 k.
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 1 0 3
%e 0 1 1 2 4
%e 0 1 0 4 4 7
%e 0 1 1 5 6 5 14
%e 0 1 0 6 10 10 14 23
%e 0 1 1 6 14 12 29 26 39
%e 0 1 0 9 16 19 40 54 46 71
%e 0 1 1 8 22 22 64 82 96 92 124
%e 0 1 0 10 26 30 82 137 144 204 176 214
%e 0 1 1 11 32 31 121 186 240 331 393 323 378
%e Row n = 6 counts the following compositions:
%e . (111111) (222) (33) (3111) (411) (6)
%e (2211) (1113) (114) (51)
%e (1122) (1221) (1311) (15)
%e (21111) (12111) (1131) (42)
%e (11112) (11211) (2112) (24)
%e (11121) (141)
%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(6,4).
%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#]]==k&]], {n,0,10},{k,0,n}]
%Y Column k = n is A003242 (anti-runs or compressed compositions).
%Y Row-sums are A011782.
%Y Same as A373951 with rows reversed.
%Y Column k = 3 is A373952.
%Y This statistic is represented by A373953, difference A373954.
%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, A333489, A373950.
%K nonn,tabl
%O 0,10
%A _Gus Wiseman_, Jun 28 2024
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