A373949 - OEIS

%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