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A374702
Number of integer compositions of n whose leaders of maximal weakly decreasing runs sum to 3. Column k = 3 of A374748.
3
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
0,4
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.
FORMULA
G.f.: x^3*(2 + x + x^2)/((1 + x + x^2)*(1 + x)*(1 - x)^3). - Andrew Howroyd, Aug 14 2024
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#, GreaterEqual]]==3&]], {n, 0, 15}]
PROG
(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
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.
Sequence in context: A248187 A190772 A130673 * A306777 A062891 A018599
KEYWORD
nonn,easy
AUTHOR
Gus Wiseman, Aug 12 2024
EXTENSIONS
a(27) onwards from Andrew Howroyd, Aug 14 2024
STATUS
approved