cp's OEIS Frontend

A333150 Number of strict compositions of n whose non-adjacent parts are strictly decreasing.

%I A333150 #11 Apr 17 2021 03:42:16
%S A333150 1,1,1,3,3,5,8,10,13,18,26,31,42,52,68,89,110,136,173,212,262,330,398,
%T A333150 487,592,720,864,1050,1262,1508,1804,2152,2550,3037,3584,4236,5011,
%U A333150 5880,6901,8095,9472,11048,12899,14996,17436,20261,23460,27128,31385,36189
%N A333150 Number of strict compositions of n whose non-adjacent parts are strictly decreasing.
%C A333150 A composition of n is a finite sequence of positive integers summing to n. It is strict if there are no repeated parts.
%H A333150 Andrew Howroyd, <a href="/A333150/b333150.txt">Table of n, a(n) for n = 0..1000</a>
%F A333150 G.f.: Sum_{k>=0} Fibonacci(k+1) * [y^k](Product_{j>=1} 1 + y*x^j). - _Andrew Howroyd_, Apr 16 2021
%e A333150 The a(1) = 1 through a(8) = 13 compositions:
%e A333150   (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)
%e A333150             (1,2)  (1,3)  (1,4)  (1,5)    (1,6)    (1,7)
%e A333150             (2,1)  (3,1)  (2,3)  (2,4)    (2,5)    (2,6)
%e A333150                           (3,2)  (4,2)    (3,4)    (3,5)
%e A333150                           (4,1)  (5,1)    (4,3)    (5,3)
%e A333150                                  (2,3,1)  (5,2)    (6,2)
%e A333150                                  (3,1,2)  (6,1)    (7,1)
%e A333150                                  (3,2,1)  (2,4,1)  (2,5,1)
%e A333150                                           (4,1,2)  (3,4,1)
%e A333150                                           (4,2,1)  (4,1,3)
%e A333150                                                    (4,3,1)
%e A333150                                                    (5,1,2)
%e A333150                                                    (5,2,1)
%e A333150 For example, (3,5,1,2) is such a composition, because the non-adjacent pairs of parts are (3,1), (3,2), (5,2), all of which are strictly decreasing.
%t A333150 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!MatchQ[#,{___,x_,__,y_,___}/;y>x]&]],{n,0,10}]
%o A333150 (PARI) seq(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); Vec(sum(k=0, n, fibonacci(k+1) * polcoef(p,k,y)))} \\ _Andrew Howroyd_, Apr 16 2021
%Y A333150 The case of permutations appears to be A000045(n + 1).
%Y A333150 Unimodal strict compositions are A072706.
%Y A333150 A version for ordered set partitions is A332872.
%Y A333150 The non-strict version is A333148.
%Y A333150 Cf. A001523, A028859, A056242, A059204, A107429, A115981, A329398, A332578, A332669, A332673, A332724, A332834, A333193.
%K A333150 nonn
%O A333150 0,4
%A A333150 _Gus Wiseman_, May 16 2020