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.

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

Offset: 0

Author

Gus Wiseman, Jun 28 2024

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    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   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  82  64  22  22   8   1   1   0
Row n = 6 counts the following compositions:
  (6)     (411)   (3111)   (33)     (222)  (111111)  .
  (51)    (114)   (1113)   (2211)
  (15)    (1311)  (1221)   (1122)
  (42)    (1131)  (12111)  (21111)
  (24)    (2112)  (11211)  (11112)
  (141)           (11121)
  (321)
  (312)
  (231)
  (213)
  (132)
  (123)
  (2121)
  (1212)
For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,2).
		

Crossrefs

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

Row-sums are A011782.

Same as A373949 with rows reversed.

Column k = 1 is A373950.

This statistic is represented by A373954, difference A373953.

A114901 counts compositions with no isolated parts.

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

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

A240085 counts compositions with no unique parts.

A333755 counts compositions by compressed length.

A373948 represents the run-compression transformation.

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

Programs

Mathematica

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