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A373951 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.

%I A373951 #6 Jun 28 2024 10:30:56
%S A373951 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 A373951 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 A373951 82,64,22,22,8,1,1,0,214,176,204,144,137,82,30,26,10,0,1,0
%N A373951 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 A373951 Triangle begins:
%e A373951     1
%e A373951     1   0
%e A373951     1   1   0
%e A373951     3   0   1   0
%e A373951     4   2   1   1   0
%e A373951     7   4   4   0   1   0
%e A373951    14   5   6   5   1   1   0
%e A373951    23  14  10  10   6   0   1   0
%e A373951    39  26  29  12  14   6   1   1   0
%e A373951    71  46  54  40  19  16   9   0   1   0
%e A373951   124  92  96  82  64  22  22   8   1   1   0
%e A373951 Row n = 6 counts the following compositions:
%e A373951   (6)     (411)   (3111)   (33)     (222)  (111111)  .
%e A373951   (51)    (114)   (1113)   (2211)
%e A373951   (15)    (1311)  (1221)   (1122)
%e A373951   (42)    (1131)  (12111)  (21111)
%e A373951   (24)    (2112)  (11211)  (11112)
%e A373951   (141)           (11121)
%e A373951   (321)
%e A373951   (312)
%e A373951   (231)
%e A373951   (213)
%e A373951   (132)
%e A373951   (123)
%e A373951   (2121)
%e A373951   (1212)
%e A373951 For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,2).
%t A373951 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#]]==n-k&]],{n,0,10},{k,0,n}]
%Y A373951 Column k = 0 is A003242 (anti-runs or compressed compositions).
%Y A373951 Row-sums are A011782.
%Y A373951 Same as A373949 with rows reversed.
%Y A373951 Column k = 1 is A373950.
%Y A373951 This statistic is represented by A373954, difference A373953.
%Y A373951 A114901 counts compositions with no isolated parts.
%Y A373951 A116861 counts partitions by compressed sum, by compressed length A116608.
%Y A373951 A124767 counts runs in standard compositions, anti-runs A333381.
%Y A373951 A240085 counts compositions with no unique parts.
%Y A373951 A333755 counts compositions by compressed length.
%Y A373951 A373948 represents the run-compression transformation.
%Y A373951 Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333382, A333489, A373952.
%K A373951 nonn,tabl
%O A373951 0,7
%A A373951 _Gus Wiseman_, Jun 28 2024