A351017 Number of binary words of length n with all distinct run-lengths.

1, 2, 2, 6, 6, 10, 22, 26, 38, 54, 114, 130, 202, 266, 386, 702, 870, 1234, 1702, 2354, 3110, 5502, 6594, 9514, 12586, 17522, 22610, 31206, 48630, 60922, 83734, 111482, 149750, 196086, 261618, 336850, 514810, 631946, 862130, 1116654, 1502982, 1916530, 2555734, 3242546

Offset: 0

Author

Gus Wiseman, Feb 07 2022

Keywords

• nonn

Examples

			The a(0) = 1 through a(6) = 22 words:
  {}  0   00   000   0000   00000   000000
      1   11   001   0001   00001   000001
               011   0111   00011   000011
               100   1000   00111   000100
               110   1110   01111   000110
               111   1111   10000   001000
                            11000   001110
                            11100   001111
                            11110   011000
                            11111   011100
                                    011111
                                    100000
                                    100011
                                    100111
                                    110000
                                    110001
                                    110111
                                    111001
                                    111011
                                    111100
                                    111110
                                    111111
		

Crossrefs

Using binary expansions instead of words gives A032020, ranked by A044813.

The version for partitions is A098859.

The complement is counted by twice A261982.

The version for compositions is A329739, for runs A351013.

For runs instead of run-lengths we have A351016, twice A351018.

The version for patterns is A351292, for runs A351200.

A000120 counts binary weight.

A001037 counts binary Lyndon words, necklaces A000031, aperiodic A027375.

A005811 counts runs in binary expansion.

A011782 counts integer compositions.

A242882 counts compositions with distinct multiplicities.

A297770 counts distinct runs in binary expansion.

A325545 counts compositions with distinct differences.

A329767 counts binary words by runs-resistance.

A351014 counts distinct runs in standard compositions.

A351204 counts partitions where every permutation has all distinct runs.

A351290 ranks compositions with all distinct runs.

Cf. A003242, A098504, A106356, A116608, A175413, A233564, A238130 or A238279, A328592, A334028, A350952.

Programs

Mathematica

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

Python

from itertools import groupby, product
def adrl(s):
    runlens = [len(list(g)) for k, g in groupby(s)]
    return len(runlens) == len(set(runlens))
def a(n):
    if n == 0: return 1
    return 2*sum(adrl("1"+"".join(w)) for w in product("01", repeat=n-1))
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 08 2022

Formula

a(n>0) = 2 * A032020(n).

Extensions

a(25)-a(32) from Michael S. Branicky, Feb 08 2022

More terms from David A. Corneth, Feb 08 2022 using data from A032020