A374702 Number of integer compositions of n whose leaders of maximal weakly decreasing runs sum to 3. Column k = 3 of A374748.

0, 0, 0, 2, 3, 6, 9, 13, 17, 23, 28, 35, 42, 50, 58, 68, 77, 88, 99, 111, 123, 137, 150, 165, 180, 196, 212, 230, 247, 266, 285, 305, 325, 347, 368, 391, 414, 438, 462, 488, 513, 540, 567, 595, 623, 653, 682, 713, 744, 776, 808, 842, 875, 910, 945, 981

Offset: 0

Author

Gus Wiseman, Aug 12 2024

Keywords

• nonn
• easy

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

			The a(0) = 0 through a(8) = 17 compositions:
  .  .  .  (3)   (31)   (32)    (33)     (322)     (332)
           (12)  (112)  (122)   (321)    (331)     (3221)
                 (121)  (311)   (1122)   (1222)    (3311)
                        (1112)  (1221)   (3211)    (11222)
                        (1121)  (3111)   (11122)   (12221)
                        (1211)  (11112)  (11221)   (32111)
                                (11121)  (12211)   (111122)
                                (11211)  (31111)   (111221)
                                (12111)  (111112)  (112211)
                                         (111121)  (122111)
                                         (111211)  (311111)
                                         (112111)  (1111112)
                                         (121111)  (1111121)
                                                   (1111211)
                                                   (1112111)
                                                   (1121111)
                                                   (1211111)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..10000
• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Crossrefs

The version for k = 2 is A004526.

The version for partitions is A069905 or A001399 (shifted).

For reversed partitions we appear to have A137719.

For length instead of sum we have A241627.

For leaders of constant runs we have A373952.

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

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

Column k = 3 of A374748.

A003242 counts anti-run compositions.

A011782 counts integer compositions.

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

A274174 counts contiguous compositions, ranks A374249.

Cf. A000009, A000041, A101271, A106356, A188900, A238343, A261982, A333213.

Programs

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,GreaterEqual]]==3&]],{n,0,15}]

PARI

seq(n)={Vec((2 + x + x^2)/((1 + x + x^2)*(1 + x)*(1 - x)^3) + O(x^(n-2)), -n-1)} \\ Andrew Howroyd, Aug 14 2024

Formula

G.f.: x^3*(2 + x + x^2)/((1 + x + x^2)*(1 + x)*(1 - x)^3). - Andrew Howroyd, Aug 14 2024

Extensions

a(27) onwards from Andrew Howroyd, Aug 14 2024