A333193 Number of compositions of n whose non-adjacent parts are strictly decreasing.

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 71, 93, 122, 158, 204, 260, 332, 419, 528, 661, 825, 1023, 1267, 1560, 1916, 2344, 2860, 3476, 4217, 5097, 6147, 7393, 8872, 10618, 12685, 15115, 17977, 21336, 25276, 29882, 35271, 41551, 48872, 57385, 67277, 78745, 92040

Offset: 0

Author

Gus Wiseman, May 18 2020

Keywords

• nonn

Examples

			The a(1) = 1 through a(7) = 15 compositions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)
       (11)  (12)  (13)   (14)   (15)    (16)
             (21)  (22)   (23)   (24)    (25)
                   (31)   (32)   (33)    (34)
                   (211)  (41)   (42)    (43)
                          (221)  (51)    (52)
                          (311)  (231)   (61)
                                 (312)   (241)
                                 (321)   (322)
                                 (411)   (331)
                                 (2211)  (412)
                                         (421)
                                         (511)
                                         (2311)
                                         (3211)
For example, (2,3,1,2) is not such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), not all of which are strictly decreasing, while (2,4,1,1) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,1), (4,1), all of which are strictly decreasing.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

A version for ordered set partitions is A332872.

The case of strict compositions is A333150.

The case of normal sequences appears to be A001045.

Unimodal compositions are A001523, with strict case A072706.

Strict compositions are A032020.

Partitions with strictly increasing run-lengths are A100471.

Partitions with strictly decreasing run-lengths are A100881.

Compositions with weakly decreasing non-adjacent parts are A333148.

Compositions with strictly increasing run-lengths are A333192.

Cf. A059204, A072707, A115981, A227038, A332834, A332836, A333191, A334966.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,,y_,_}/;y>=x]&]],{n,0,15}]

PARI

\\ p is all, q is those ending in an unreversed singleton.
seq(n)={my(q=O(x*x^n), p=1+q); for(k=1, n, [p,q] = [p*(1 + x^k + x^(2*k)) + q*x^k, q + p*x^k] ); Vec(p)} \\ Andrew Howroyd, Apr 17 2021

Extensions

Terms a(21) and beyond from Andrew Howroyd, Apr 17 2021