A374517 Number of integer compositions of n whose leaders of anti-runs are identical.

1, 1, 2, 4, 7, 13, 25, 46, 85, 160, 301, 561, 1056, 1984, 3730, 7037, 13273, 25056, 47382, 89666, 169833, 322038, 611128, 1160660, 2206219, 4196730, 7988731, 15217557, 29005987, 55321015, 105570219, 201569648, 385059094, 735929616, 1407145439, 2691681402

Offset: 0

Author

Gus Wiseman, Aug 01 2024

Keywords

• nonn

Comments

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

Examples

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (1111)  (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (1211)
                                (11111)
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 0..100
• Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Crossrefs

For partitions instead of compositions we have A034296 or A115029.

These compositions have ranks A374519.

The complement is counted by A374640.

Other types of runs (instead of anti-):

- For leaders of identical runs we have A000005 for n > 0, ranks A272919.

- For leaders of weakly increasing runs we have A374631, ranks A374633.

- For leaders of strictly increasing runs we have A374686, ranks A374685.

- For leaders of weakly decreasing runs we have A374742, ranks A374741.

- For leaders of strictly decreasing runs we have A374760, ranks A374759.

Other types of run-leaders (instead of identical):

- For distinct leaders we have A374518.

- For weakly increasing leaders we have A374681.

- For strictly increasing leaders we have A374679.

- For weakly decreasing leaders we have A374682.

- For strictly decreasing leaders we have A374680.

A003242 counts anti-runs, ranks A333489.

A106356 counts compositions by number of maximal anti-runs.

A238279 counts compositions by number of maximal runs

A238424 counts partitions whose first differences are an anti-run.

A274174 counts contiguous compositions, ranks A374249.

Cf. A188920, A189076, A238343, A333213, A373949, A374515, A374518.

Programs

Mathematica

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

PARI

C_x(N) = {my(g =1/(1 - sum(k=1, N, x^k/(1+x^k))));g}
A_x(i,N) = {my(x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2);f}
B_x(i,j,N) = {my(x='x+O('x^N), f=C_x(N)*x^(i+j)/((1+x^i)*(1+x^j)));f}
D_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,-1+sum(j=0,N-i, A_x(i,N)^j)*(1-B_x(i,i,N)+sum(k=1,N-i,B_x(i,k,N)))));Vec(f)}
D_x(30) \\ John Tyler Rascoe, Aug 16 2024

Formula

G.f.: 1 + Sum_{i>0} (-1 + Sum_{j>=0} (A(i,x)^j)*(1 + Sum_{k>0, k<>i} (B(i,k,x)))) where A(i,x) = (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2, B(i,k,x) = C(x)*x^(i+k)/((1+x^i)*(1+x^k)), and C(x) is the g.f. for A003242. - John Tyler Rascoe, Aug 16 2024

Extensions

a(26) onwards from John Tyler Rascoe, Aug 16 2024