A374766 - OEIS

A374766

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

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 3, 5, 0, 0, 0, 1, 8, 7, 0, 0, 0, 1, 3, 17, 11, 0, 0, 0, 0, 4, 10, 35, 15, 0, 0, 0, 0, 1, 12, 28, 65, 22, 0, 0, 0, 0, 1, 6, 31, 70, 118, 30, 0, 0, 0, 0, 1, 3, 22, 78, 163, 203, 42, 0, 0, 0, 0, 0, 4, 13, 69, 186, 354, 342, 56

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OFFSET

0,6

COMMENTS

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

Are the column-sums finite?

LINKS

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

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

EXAMPLE

Triangle begins:

0 1

0 0 2

0 0 1 3

0 0 0 3 5

0 0 0 1 8 7

0 0 0 1 3 17 11

0 0 0 0 4 10 35 15

0 0 0 0 1 12 28 65 22

0 0 0 0 1 6 31 70 118 30

0 0 0 0 1 3 22 78 163 203 42

0 0 0 0 0 4 13 69 186 354 342 56

Row n = 6 counts the following compositions:

. . . (321) (42) (51) (6)

(132) (411) (15)

(2121) (141) (24)

(312) (114)

(231) (33)

(213) (123)

(3111) (1113)

(1311) (222)

(1131) (1122)

(2211) (11112)

(2112) (111111)

(1221)

(1212)

(21111)

(12111)

(11211)

(11121)

MATHEMATICA

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

CROSSREFS

Column n = k is A000041.

Row-sums are A011782.

For length instead of sum we have A333213.

The corresponding rank statistic is A374758, row-sums of A374757.

For identical leaders we have A374760, ranks A374759.

For distinct leaders we have A374761, ranks A374767.

Other types of runs (instead of strictly decreasing):

- For leaders of identical 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 weakly decreasing runs we have A374748.

A003242 counts anti-run compositions.

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

A335456 counts patterns matched by compositions.

Cf. A106356, A238343, A261982, A274174, A374517, A374518, A374687.

Sequence in context: A258850 A182114 A122950 * A116489 A166373 A202451

Adjacent sequences: A374763 A374764 A374765 * A374767 A374768 A374769

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Aug 02 2024

STATUS

approved