A157656 Maximal possible number of states in a minimal deterministic automaton, equivalent to an n-state nondeterministic automaton over 1-symbol alphabet.

2, 3, 6, 11, 18, 27

Offset: 1

Author

Max Alekseyev, Mar 03 2009

Keywords

• nonn
• hard
• more

Comments

Alternative definition: consider a labyrinth consisting of n rooms, one designated as the "start room", connected by a number of one-way corridors. Let R(k) be a set of all rooms that can be reached from the start room after passing through exactly k corridors. We need to construct a labyrinth with the maximal number of distinct R(k), i.e., a set { R(0), R(1), R(2), ... } (that is actually a finite set) must be of the largest possible size. This size is a(n).

For small n, a(n)=A059100(n-1) which corresponds to a labyrinth 1 -> 2 -> 3 -> ... -> n -> 1, n -> 2 with the start room "1".

For large n, a(n) is different from A059100(n-1). In particular, for n=29, there is a labyrinth of the following shape: there are five directed corridors from the start room to five other rooms that belong to disjoint directed cycles of length 2, 3, 5, 7, and 11 respectively (note that 29 = 1+2+3+5+7+11). It gives 1+2*3*5*7*11=2311 distinct R(k)'s, implying that a(29)>=2311>A059100(28).

Conjecture: a(n)=A059100(n-1) holds only for all n<20 as well as n=22 and n=23. (Rustem Aidagulov)

In general, a(n) >= A000793(n-1)+1. The strategy is to partition n-1 into coprime numbers with maximum product and create a cycle for each, then add one more starting node which is connected to all cycles. This also implies that a(n) is Omega(e^sqrt(n log n)). - Martín Muñoz, Dec 10 2023

Links

• Author?, Discussion of the problem (in Russian)

Crossrefs

Cf. A059100, A000793.