A341617 Repair factors for Stirling numbers of the second kind.

1, 1, 2, 6, 12, 60, 30, 210, 840, 2520, 1260, 13860, 13860, 180180, 90090, 30030, 240240, 4084080, 6126120, 116396280, 58198140, 58198140

Offset: 1

Author

Thomas Ward, Feb 16 2021

Keywords

• nonn
• more

Comments

a(k) is the smallest number with the property that the sequence (a(k)*S(n+k-1, k))_{n>=1}, viewed as a sequence in n, counts the periodic points of some map. Here S(n, k) denotes the Stirling numbers of the second kind, so S(n, k) is the number of ways to partition a set of n elements into k nonempty subsets.

Because these numbers relate to properties of the Stirling number sequence S(n, k) viewed as a sequence in n, it is natural to think of these values as indexed by k.

The values generated by a finite calculation are inherently empirical, calculated as the least common multiple of the first 3000 terms and then checked for the repair property for period up to 50,000. For smaller values of k (those listed here) various ad hoc arguments can be used to check the empirical candidate values.

The empirical values can be calculated using a simple PARI code, and this only gives a possible candidate value for a(k). If that value happens to be (k-1)! then it is correct. If it is a strict divisor of (k-1)! then further checks are needed for each prime in the quotient (candidate value)/(k-1)!. The potentially unbounded nature of the calculation required would make a proof of a closed formula particularly interesting.

(EMPIRICAL PARI CODE TO GENERATE CANDIDATE VALUES) for(k=1, 40, a=vector(3000, n, stirling(n+k-1, k, 2)); b=vector(length(a), n, (1/n)*sumdiv(n, d, moebius(n/d)*a[d])); print1(denominator(b)", "))

Examples

			The statement that a(3) = 2 means that the sequence of Stirling numbers S_3 = (1, 6, 25, 90, ...) (that is, the sequence A000392 with an offset of 3) does not have the property of counting periodic points for some map, but does have this property after multiplication by 2 (which gives A028243 with an offset of 2), and 2 is the smallest integer with this property. This specific value is immediately known to be exact, because a(3) divides (3-1)! = 2.
		

Links

• P. Miska and T. Ward, Stirling numbers and periodic points, arXiv:2102.07561 [math.NT], 2021.

Crossrefs

Cf. A000392, A028243, A008277, A341991.

Formula

It is known that a(k) divides (k-1)!.

There is an explicit formula for a(k), but it involves an in principle infinite calculation as follows: compute the set of rational numbers {(1/n) Sum_{d|n} mu(n/d)*S(d+k-1, k): n>=1}, and then define a(k) to be the least common multiple of the denominators of that (possibly infinite) set of rational numbers. Here mu denotes the classical Möbius function, and the sum is taken over the divisors d of n. Strictly speaking, any calculation only gives candidate values which are initially only known to be a factor of the real value, which in turn is a factor of (k-1)!. For small values of k listed above ad hoc arguments are needed to check the candidate values evaluated as the least common multiple of the denominators of the first 3000 terms.