A373949 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.

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, 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, 22, 64, 82, 96, 92, 124, 0, 1, 0, 10, 26, 30, 82, 137, 144, 204, 176, 214

Offset

0,10

Links

Table of n, a(n) for n=0..77.

Example

Triangle begins:
   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  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  22  64  82  96  92 124
   0   1   0  10  26  30  82 137 144 204 176 214
   0   1   1  11  32  31 121 186 240 331 393 323 378
Row n = 6 counts the following compositions:
  .  (111111)  (222)  (33)     (3111)   (411)   (6)
                      (2211)   (1113)   (114)   (51)
                      (1122)   (1221)   (1311)  (15)
                      (21111)  (12111)  (1131)  (42)
                      (11112)  (11211)  (2112)  (24)
                               (11121)          (141)
                                                (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(6,4).

Mathematica

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

Crossrefs

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

Row-sums are A011782.

Same as A373951 with rows reversed.

Column k = 3 is A373952.

This statistic is represented by A373953, difference A373954.

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, A333489, A373950.

Sequence in context: A131802 A069023 A275336 * A091614 A350829 A249767

Adjacent sequences: A373946 A373947 A373948 * A373950 A373951 A373952

Keyword

nonn,tabl

Author

Gus Wiseman, Jun 28 2024

Status

approved