

A069916


Number of logconcave compositions (ordered partitions) of n.


34



1, 1, 2, 4, 6, 9, 14, 20, 26, 36, 47, 60, 80, 102, 127, 159, 194, 236, 291, 355, 425, 514, 611, 718, 856, 1009, 1182, 1381, 1605, 1861, 2156, 2496, 2873, 3299, 3778, 4301, 4902, 5574, 6325, 7176, 8116, 9152, 10317, 11610, 13028, 14611, 16354, 18259, 20365
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,3


COMMENTS

These are compositions with weakly decreasing first quotients, where the first quotients of a sequence are defined as if the sequence were an increasing divisor chain, so for example the first quotients of (6,3,1) are (1/2,1/3).  Gus Wiseman, Mar 16 2021


LINKS



EXAMPLE

Out of the 8 compositions of 4, only 2+1+1 and 1+1+2 are not logconcave, so a(4)=6.
The a(1) = 1 through a(6) = 14 compositions:
(1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(21) (22) (23) (24)
(111) (31) (32) (33)
(121) (41) (42)
(1111) (122) (51)
(131) (123)
(221) (132)
(11111) (141)
(222)
(231)
(321)
(1221)
(111111)
(End)


MATHEMATICA

(* This program is not suitable for computing a large number of terms *)
compos[n_] := Permutations /@ IntegerPartitions[n] // Flatten[#, 1]&;
logConcaveQ[p_] := And @@ Table[p[[i]]^2 >= p[[i1]]*p[[i+1]], {i, 2, Length[p]1}]; a[n_] := Count[compos[n], p_?logConcaveQ]; Table[an = a[n]; Print["a(", n, ") = ", an]; a[n], {n, 0, 25}] (* JeanFrançois Alcover, Feb 29 2016 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], GreaterEqual@@Divide@@@Reverse/@Partition[#, 2, 1]&]], {n, 0, 15}] (* Gus Wiseman, Mar 15 2021 *)


PROG

(Sage) def A069916(n) : return sum(all(p[i]^2 >= p[i1] * p[i+1] for i in range(1, len(p)1)) for p in Compositions(n)) # Eric M. Schmidt, Sep 29 2013


CROSSREFS

The version for differences instead of quotients is A070211.
A000005 counts constant compositions.
A000009 counts strictly increasing (or strictly decreasing) compositions.
A000041 counts weakly increasing (or weakly decreasing) compositions.
A002843 counts compositions with adjacent parts x <= 2y.
A003238 counts chains of divisors summing to n1, with strict case A122651.
A003242 counts antirun compositions.
A074206 counts ordered factorizations.
A167865 counts strict chains of divisors summing to n.


KEYWORD

nonn,nice


AUTHOR



STATUS

approved



