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A325548
Number of compositions of n with strictly decreasing differences.
12
1, 1, 2, 3, 5, 8, 10, 13, 19, 23, 29, 38, 46, 55, 69, 80, 96, 115, 132, 154, 183, 207, 238, 276, 314, 356, 405, 455, 513, 579, 647, 724, 809, 897, 998, 1107, 1225, 1350, 1486, 1639, 1805, 1973, 2166, 2374, 2586, 2824, 3084, 3346, 3646, 3964, 4286, 4655, 5047
OFFSET
0,3
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (3,1,2) are (-2,1).
EXAMPLE
The a(1) = 1 through a(8) = 19 compositions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (12) (13) (14) (15) (16) (17)
(21) (22) (23) (24) (25) (26)
(31) (32) (33) (34) (35)
(121) (41) (42) (43) (44)
(122) (51) (52) (53)
(131) (132) (61) (62)
(221) (141) (133) (71)
(231) (142) (134)
(1221) (151) (143)
(232) (152)
(241) (161)
(331) (233)
(242)
(251)
(332)
(341)
(431)
(1331)
MAPLE
b:= proc(n, l, d) option remember; `if`(n=0, 1, add(`if`(l=0 or
j-l<d, b(n-j, j, `if`(l=0, infinity, j-l)), 0), j=1..n))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..52); # Alois P. Heinz, Jan 27 2024
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Greater@@Differences[#]&]], {n, 0, 15}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 10 2019
EXTENSIONS
a(26)-a(44) from Lars Blomberg, May 30 2019
a(45)-a(52) from Alois P. Heinz, Jan 27 2024
STATUS
approved