A374746 Number of integer compositions of n whose leaders of weakly decreasing runs are strictly decreasing.

1, 1, 2, 3, 5, 7, 12, 18, 31, 51, 86, 143, 241, 397, 657, 1082, 1771, 2889, 4697, 7605, 12269, 19720, 31580, 50412, 80205, 127208, 201149, 317171, 498717, 782076, 1223230, 1908381, 2969950, 4610949, 7141972, 11037276, 17019617, 26188490, 40213388, 61624824

Offset: 0

Author

Gus Wiseman, Jul 26 2024

Keywords

• nonn

Comments

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

Examples

			The a(0) = 1 through a(7) = 18 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (312)     (331)
                                (11111)  (321)     (412)
                                         (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (3112)
                                         (111111)  (3121)
                                                   (3211)
                                                   (4111)
                                                   (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

Ranked by positions of strictly decreasing rows in A374740, opp. A374629.

Types of runs (instead of weakly decreasing):

- For leaders of identical runs we have A000041.

- For leaders of weakly increasing runs we have A188920.

- For leaders of anti-runs we have A374680.

- For leaders of strictly increasing runs we have A374689.

- For leaders of strictly decreasing runs we have A374763.

Types of run-leaders (instead of strictly decreasing):

- For weakly increasing leaders we appear to have A188900.

- For identical leaders we have A374742.

- For distinct leaders we have A374743, ranks A374701.

- For strictly increasing leaders we have opposite A374634.

- For weakly decreasing leaders we have A374747.

A011782 counts compositions.

A238130, A238279, A333755 count compositions by number of runs.

A335456 counts patterns matched by compositions.

A373949 counts compositions by run-compressed sum, opposite A373951.

A374748 counts compositions by sum of leaders of weakly decreasing runs.

Cf. A000009, A003242, A106356, A189076, A238343, A261982, A333213, A358836, A374632, A374635, A374741.

Programs

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,GreaterEqual]&]],{n,0,15}]

PARI

seq(n)={my(A=O(x*x^n), p=1+A, q=p, r=p); for(k=1, n\2, r += x^k*q/(1-x^k); p /= 1 - x^k; q *= (1 - x^k/(1-x^k) + x^k*p)/(1-x^k) );  Vec(r + x^(n\2+1)*q/(1-x))} \\ Andrew Howroyd, Dec 30 2024

Formula

G.f.: Sum_{k>=0} x^k*Q(k,x)/(1 - x^k) where Q(0,x) = 1 and Q(k,x) = Q(k-1,x) * (1 - x^k/(1 - x^k) + x^k*Product_{j=1..k} (1 - x^j))/(1 - x^k) for k > 0. - Andrew Howroyd, Dec 30 2024

Extensions

a(24)-a(39) from Alois P. Heinz, Jul 26 2024