A374748 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly decreasing runs sum to k.

1, 0, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 3, 2, 0, 1, 2, 6, 4, 3, 0, 1, 3, 9, 8, 7, 4, 0, 1, 3, 13, 15, 16, 11, 5, 0, 1, 4, 17, 24, 32, 28, 16, 6, 0, 1, 4, 23, 36, 58, 58, 44, 24, 8, 0, 1, 5, 28, 52, 96, 115, 100, 71, 34, 10, 0, 1, 5, 35, 72, 151, 203, 211, 176, 109, 49, 12

Offset: 0

Author

Gus Wiseman, Jul 26 2024

Keywords

• nonn
• tabl

Comments

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

Examples

			Triangle begins:
   1
   0   1
   0   1   1
   0   1   1   2
   0   1   2   3   2
   0   1   2   6   4   3
   0   1   3   9   8   7   4
   0   1   3  13  15  16  11   5
   0   1   4  17  24  32  28  16   6
   0   1   4  23  36  58  58  44  24   8
   0   1   5  28  52  96 115 100  71  34  10
   0   1   5  35  72 151 203 211 176 109  49  12
Row n = 6 counts the following compositions:
  .  (111111)  (222)    (33)     (42)    (51)    (6)
               (2211)   (321)    (411)   (141)   (15)
               (21111)  (3111)   (132)   (114)   (24)
                        (1221)   (1311)  (312)   (123)
                        (1122)   (1131)  (231)
                        (12111)  (1113)  (213)
                        (11211)  (2121)  (1212)
                        (11121)  (2112)
                        (11112)
		

Links

• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

Column n = k is A000009.

Column k = 2 is A004526.

Row-sums are A011782.

For length instead of sum we have A238343.

The opposite rank statistic is A374630, row-sums of A374629.

Column k = 3 is A374702.

The center n = 2k is A374703.

The corresponding rank statistic is A374741 row-sums of A374740.

Types of runs (instead of weakly decreasing):

- For leaders of constant runs we have A373949.

- For leaders of anti-runs we have A374521.

- For leaders of weakly increasing runs we have A374637.

- For leaders of strictly increasing runs we have A374700.

- For leaders of strictly decreasing runs we have A374766.

Types of run-leaders:

- For weakly increasing leaders we appear to have A188900.

- For identical leaders we have A374742, ranks A374744.

- For distinct leaders we have A374743, ranks A374701.

- For strictly decreasing leaders we have A374746.

- For weakly decreasing leaders we have A374747.

A003242 counts anti-run compositions.

A238130, A238279, A333755 count compositions by number of runs.

A274174 counts contiguous compositions, ranks A374249.

A335456 counts patterns matched by compositions.

A335548 counts non-contiguous compositions, ranks A374253.

Cf. A000041, A106356, A124766, A261982, A333213, A374251, A374761.

Programs

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,GreaterEqual]]==k&]],{n,0,15},{k,0,n}]