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A094913 Maximal number of distinct nonempty substrings of any binary string of length n. 4
0, 1, 3, 5, 8, 12, 16, 21, 27, 34, 42, 50, 59, 69, 80, 92, 105, 119, 134, 150, 166, 183, 201, 220, 240, 261, 283, 306, 330 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

It would be more natural to include the empty substring, which would result in the sequence b(n)=a(n)+1; all values computed so far confirm the conjecture that a(n)+1 = A006697(n). - M. F. Hasler, Dec 17 2007

In addition to the example given, computation finds 17 other binary strings of length 6 which contain the maximal number a(6)=16 of distinct substrings. Interestingly, each of the 18 such instances gives the same numbers of substrings of each possible length, in this case 2,4,4,3,2,1 substrings of lengths 1 through 6, respectively.

Calculations suggest that, for any n>0, binary strings of length n exist such that the number of distinct substrings of length k, 1<=k<=n, is as large as possible consistent with basic counting principles, i.e., n-k+1 substrings of length k starting at each of the n-k+1 possible starting positions, subject to the condition, however, that there cannot be more than 2^k distinct binary strings of length k. This suggests the following conjecture. Conjecture: The maximum number a(n) of distinct substrings of a binary string of length n is given by a(n) = Sum_{k=1..n} t(k) where t(k) is the smaller of 2^k and n-k+1.

Conjecture: a(n) = A006697(n)-1. - Vladeta Jovovic, Sep 19 2005

Conjecture: a(n) = 2^(m+1)-2 + (n-m)(n-m+1)/2, where m = floor(n+1-LambertW( 2^(n+1) * log(2) ) / log(2) ) = integer part of the solution to 2^x = n+1-x. This is based on the reasoning that you can construct the word of length n so that it contains the maximal possible number, max( n-k+1, 2^k ), of different substrings of length k. - M. F. Hasler, Dec 17 2007

From Peter Pein (petsie(AT)dordos.net), Jan 18 2008: (Start)

The following heuristic seems to work for computing the maximal number of distinct substrings of a binary string of length n.

Start with the empty list and at each step try to insert a zero or a one at each possible position. Then pick the first list with the maximal number of sublists and start over.

Say we have had {0,0,1,1,0} as one of the lists with the maximal number of sublists. Then my candidates for the next test are:

With[{lastbest = {0, 0, 1, 1, 0}}, Union[Flatten[ Outer[Insert[lastbest, #2, #1] &, Range[1 + Length[lastbest]], {0, 1}, 1], 1]]]

{{0, 0, 0, 1, 1, 0}, {0, 0, 1, 0, 1, 0}, {0, 0, 1, 1, 0, 0}, {0, 0, 1, 1, 0, 1}, {0, 0, 1, 1, 1, 0}, {0, 1, 0, 1, 1, 0}, {1, 0, 0, 1, 1, 0}}

See http://freenet-homepage.de/Peter_Berlin/Mathe/heuristicA094913.nb for the Mathematica notebook with the complete (simple) algorithm. There's a screenshot too (same URL but with .png instead of .nb).

If this works correctly, the first 100 values of A094913 are calculated in 30 secs.

(End) [See also the remarks in the Fuller link in A134457.]

From Jon Perry, Nov 16 2010: (Start)

Conjecture: column sums of:

1 3 5 7 9  11 13 ...

      1 3   5  7 ...

               1 ...

....................

--------------------

1 3 5 8 12 16 21 ... (End)

LINKS

Table of n, a(n) for n=0..28.

The history of this problem is given in a comment in the Electronic Journal of Combinatorics. Some of the conjectures above are proven in the papers cited.

J.-P. Allouche, J. Shallit, On the subword complexity of the fixed point of a -> aab, b -> b, and generalizations, arXiv preprint arXiv:1605.02361 [math.CO], 2016.

Abraham Flaxman, Aram W. Harrow, Gregory B. Sorkin, Strings with Maximally Many Distinct Subsequences and Substrings, Electronic J. Combinatorics 11 (1) (2004), Paper R8.

Antal Iványi, On the d-complexity of words, Ann. Univ. Sci. Budapest. Sect. Comput. 8 (1987), 69-90 (1988).

Jeffrey Shallit, On the maximum number of distinct factors in a binary string, Graphs Combin. 9 (1993), no. 2, 197-200.

EXAMPLE

The binary string 000110 of length 6 contains the 16 distinct substrings 0, 1, 00, 01, 11, 10, 000, 001, 011, 110, 0001, 0011, 0110, 00011, 00110, 000110 and a computer search shows that no other binary string of length 6 contains more than 16. Thus a(6)=16.

G.f. = x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 16*x^6 + 21*x^7 + 27*x^8 + 34*x^9 + ...

MATHEMATICA

A103354[n_] := Floor[ FullSimplify[ ProductLog[ 2^n*Log[2]]/Log[2]]]; Accumulate[ Table[ A103354[n], {n, 1, 29}]] - 1 (* Jean-François Alcover, Dec 13 2011, after M. F. Hasler *)

CROSSREFS

Cf. A006697, A134457, A134466.

Sequence in context: A194180 A000212 A183139 * A265060 A320849 A255981

Adjacent sequences:  A094910 A094911 A094912 * A094914 A094915 A094916

KEYWORD

nonn,nice

AUTHOR

John W. Layman, Jun 17 2004

EXTENSIONS

a(19)-a(28) from David W. Wilson, Dec 17 2007

More references (some proving some conjectures) added by Jeffrey Shallit, Nov 21 2015

STATUS

approved

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Last modified September 20 07:21 EDT 2020. Contains 337264 sequences. (Running on oeis4.)