A374763 Number of integer compositions of n whose leaders of strictly decreasing runs are themselves strictly decreasing.

1, 1, 1, 2, 3, 4, 6, 10, 15, 22, 32, 47, 71, 106, 156, 227, 328, 473, 683, 986, 1421, 2040, 2916, 4149, 5882, 8314, 11727, 16515, 23221, 32593, 45655, 63810, 88979, 123789, 171838, 238055, 329187, 454451, 626412, 862164, 1184917, 1626124, 2228324, 3048982, 4165640, 5682847

Offset: 0

Author

Gus Wiseman, Jul 30 2024

Keywords

• nonn

Comments

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

Examples

			The composition (3,1,2,1,1) has strictly decreasing runs ((3,1),(2,1),(1)), with leaders (3,2,1), so is counted under a(8).
The a(0) = 1 through a(8) = 15 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)
                (21)  (31)   (32)   (42)   (43)    (53)
                      (211)  (41)   (51)   (52)    (62)
                             (311)  (312)  (61)    (71)
                                    (321)  (322)   (413)
                                    (411)  (412)   (422)
                                           (421)   (431)
                                           (511)   (512)
                                           (3121)  (521)
                                           (3211)  (611)
                                                   (3212)
                                                   (3221)
                                                   (4121)
                                                   (4211)
                                                   (31211)
		

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

The opposite version is A374688.

The weak version is A374747.

For partitions instead of compositions we have A375133.

Other types of runs (instead of strictly decreasing):

- For leaders of identical runs we have A000041.

- For leaders of weakly increasing runs we appear to have A188920.

- For leaders of anti-runs we have A374680.

- For leaders of strictly increasing runs we have A374689.

- For leaders of weakly decreasing runs we have A374746.

Other types of run-leaders (instead of strictly decreasing):

- For identical leaders we have A374760, ranks A374759.

- For distinct leaders we have A374761, ranks A374767.

- For strictly increasing leaders we have A374762.

- For weakly increasing leaders we have A374764.

- For weakly decreasing leaders we have A374765.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

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

A274174 counts contiguous compositions, ranks A374249.

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

Cf. A000009, A106356, A188900, A238343, A261982, A333213, A374518, A374632, A374635, A374687, A374742, A374743.

Programs

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Greater@@First/@Split[#,Greater]&]],{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; p *= 1 + x^k; q *= 1 + x^k*p); 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) where Q(0,x) = 1 and Q(k,x) = Q(k-1,x) * (1 + x^k*Product_{j=1..k} (1 + x^j)) for k > 0. - Andrew Howroyd, Dec 30 2024

Extensions

a(24) onwards from Andrew Howroyd, Dec 30 2024