A351018 Number of integer compositions of n with all distinct even-indexed parts and all distinct odd-indexed parts.

1, 1, 2, 3, 6, 9, 18, 27, 46, 77, 122, 191, 326, 497, 786, 1207, 1942, 2905, 4498, 6703, 10574, 15597, 23754, 35043, 52422, 78369, 115522, 169499, 248150, 360521, 532466, 768275, 1116126, 1606669, 2314426, 3301879, 4777078, 6772657, 9677138, 13688079, 19406214

Offset

0,3

Comments

Also the number of binary words of length n starting with 1 and having all distinct runs (ranked by A175413, counted by A351016).

Links

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

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Formula

a(n>0) = A351016(n)/2.

G.f.: Sum_{k>=0} floor(k/2)! * ceiling(k/2)! * ([y^floor(k/2)] P(x,y)) * ([y^ceiling(k/2)] P(x,y)), where P(x,y) = Product_{k>=1} 1 + y*x^k. - Andrew Howroyd, Feb 11 2022

Example

The a(1) = 1 through a(6) = 18 compositions:
  (1)  (2)    (3)    (4)      (5)      (6)
       (1,1)  (1,2)  (1,3)    (1,4)    (1,5)
              (2,1)  (2,2)    (2,3)    (2,4)
                     (3,1)    (3,2)    (3,3)
                     (1,1,2)  (4,1)    (4,2)
                     (2,1,1)  (1,1,3)  (5,1)
                              (1,2,2)  (1,1,4)
                              (2,2,1)  (1,2,3)
                              (3,1,1)  (1,3,2)
                                       (2,1,3)
                                       (2,3,1)
                                       (3,1,2)
                                       (3,2,1)
                                       (4,1,1)
                                       (1,1,2,2)
                                       (1,2,2,1)
                                       (2,1,1,2)
                                       (2,2,1,1)

Mathematica

Table[Length[Select[Tuples[{0, 1}, n], #=={}||First[#]==1&&UnsameQ@@Split[#]&]], {n, 0, 10}]

Prog

(PARI) P(n)=prod(k=1, n, 1 + y*x^k + O(x*x^n));
seq(n)=my(p=P(n)); Vec(sum(k=0, n, polcoef(p, k\2, y)*(k\2)!*polcoef(p, (k+1)\2, y)*((k+1)\2)!)) \\ Andrew Howroyd, Feb 11 2022

Crossrefs

The case of partitions is A000726.

The version for run-lengths instead of runs is A032020.

These words are ranked by A175413.

A005811 counts runs in binary expansion.

A011782 counts integer compositions.

A044813 lists numbers whose binary expansion has distinct run-lengths.

A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.

A116608 counts compositions by number of distinct parts.

A238130 and A238279 count compositions by number of runs.

A242882 counts compositions with distinct multiplicities.

A297770 counts distinct runs in binary expansion.

A325545 counts compositions with distinct differences.

A329738 counts compositions with equal run-lengths.

A329744 counts compositions by runs-resistance.

A351014 counts distinct runs in standard compositions.

Counting words with all distinct runs:

- A351013 = compositions, for run-lengths A329739, ranked by A351290.

- A351016 = binary words, for run-lengths A351017.

- A351200 = patterns, for run-lengths A351292.

- A351202 = permutations of prime factors.

Cf. A003242, A025047, A098504, A098859, A106356, A212322, A328592, A329740, A334028, A349054, A350952, A351205.

Sequence in context: A124879 A261661 A062865 * A180684 A206100 A178940

Adjacent sequences: A351015 A351016 A351017 * A351019 A351020 A351021

Keyword

nonn

Author

Gus Wiseman, Feb 09 2022

Extensions

Terms a(21) and beyond from Andrew Howroyd, Feb 11 2022

Status

approved