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Number of compositions of n with strictly decreasing differences.
12

%I #17 Jan 27 2024 15:31:52

%S 1,1,2,3,5,8,10,13,19,23,29,38,46,55,69,80,96,115,132,154,183,207,238,

%T 276,314,356,405,455,513,579,647,724,809,897,998,1107,1225,1350,1486,

%U 1639,1805,1973,2166,2374,2586,2824,3084,3346,3646,3964,4286,4655,5047

%N Number of compositions of n with strictly decreasing differences.

%C A composition of n is a finite sequence of positive integers summing to n.

%C 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).

%H Alois P. Heinz, <a href="/A325548/b325548.txt">Table of n, a(n) for n = 0..400</a>

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e The a(1) = 1 through a(8) = 19 compositions:

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (11) (12) (13) (14) (15) (16) (17)

%e (21) (22) (23) (24) (25) (26)

%e (31) (32) (33) (34) (35)

%e (121) (41) (42) (43) (44)

%e (122) (51) (52) (53)

%e (131) (132) (61) (62)

%e (221) (141) (133) (71)

%e (231) (142) (134)

%e (1221) (151) (143)

%e (232) (152)

%e (241) (161)

%e (331) (233)

%e (242)

%e (251)

%e (332)

%e (341)

%e (431)

%e (1331)

%p b:= proc(n, l, d) option remember; `if`(n=0, 1, add(`if`(l=0 or

%p j-l<d, b(n-j, j, `if`(l=0, infinity, j-l)), 0), j=1..n))

%p end:

%p a:= n-> b(n, 0$2):

%p seq(a(n), n=0..52); # _Alois P. Heinz_, Jan 27 2024

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Greater@@Differences[#]&]],{n,0,15}]

%Y Cf. A011782, A000740, A008965, A070211, A175342, A179254, A320470, A325457, A325545, A325546, A325547, A325552, A325557.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 10 2019

%E a(26)-a(44) from _Lars Blomberg_, May 30 2019

%E a(45)-a(52) from _Alois P. Heinz_, Jan 27 2024