COMMENT on A114717. Increasing compositeness?

[Antti Karttunen]

Harris, Mitchell A., Ph.D. wrote:

>Prompted by Antti's seqfan post (and Frank's reply), I noticed a great absence
>of references to linear extensions in the OEIS, and also some sequences for
>which I calculated values a few years ago but inexplicably did not submit. Here
>follows an attempt to fix that. First are comments to add to existing sequences,
>then come a handful of new sequences, and lastly entries for the index. If there
>are other indexable poset families that should be listed, or there are better
>comments to be made about these sequences, please inform me. 
>
>  
>
...

Thanks for saving me some submission work, and also for the index 
extensions!
However, I have also a few comments to this:
(CC: SeqFan, for all to ponder...)

>
>%I A114717 
>%S A114717 1,1,1,1,1,2,1,1,1,2,1,5,1,2,2,1,1,5,1,5,2,2,1,14,1,2,1,5,1,48,1,1,2,
>%T A114717 2,2,42,1,2,2,14,1,48,1,5,5,2,1,42,1,5,2,5,1,14,2,14,2,2,1,2452,1,2,5,
>%U A114717 1,2,48,1,5,2,48,1,462,1,2,5,5,2,48,1,42,1,2,1,2452,2,2,2,14,1,2452,2
>%N A114717 Number of linear extensions of the divisor lattice of n.
>%D A114717 Stanley, R., Enumerative Combinatorics, Vol.2, Proposition 7.10.3 and
>Vol 1, Sec 3.5 Chains in Distributive Lattices.
>%D A114717 Pruesse, Gary and Ruskey, Frank, Generating linear extensions fast,
>SIAM J.Comput.23 (1994), no. 2, 373-386.
>%D A114717 Brightwell, Graham, and Winkler, Peter, Counting linear extensions.
>Order 8 (1991), no. 3, 225-242.
>%Y A114717 Cf. A060854, A114714, A114715, A114716
>%A A114717 Mitch Harris (Harris.Mitchell (AT) mgh.harvard.edu) and Antti
>Karttunen (His-Firstname.His-Surname(AT)iki.fi), Dec 27, 2005
>%O A114717 1,6
>%K A114717 nonn,hard
>%C A114717 Notice that only the powers of the primes determine a(n), so a(12) =
>a(75) = 5.
>%C A114717 For prime powers, the lattice is a chain, so there is 1 linear
>extension.
>%C A114717 For n = p1^r1 * p2^r2, the lattice is a grid (r1+1)*(r2+1), whose
>linear extensions are counted by ((r1+1)(r2+1))!/Product[(r1+k)!/k!,{k,0,r2}].
>Cf. A060854.
>%C A114717 a(p^1*q^n) = A000108(n+1), the Catalan numbers.
>
>  
>
Comment and example lines to add:

%C A114717 In other words gives the number of ways to arrange the 
divisors of n in such a way, that no divisor has any of its own divisors 
following after it.
%e A114717 E.g. for 12, the following five arrangements are possible: 
1,2,3,4,6,12; 1,2,3,6,4,12; 1,2,4,3,6,12; 1,3,2,4,6,12 and 1,3,2,6,4,12. 
But e.g. 1,2,6,4,3,12 is not possible, because 3 divides 6 but follows 
after it. Thus a(12)=5.

(and by the way, I nowadays prefer this way to put my e-mail address: 
Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com) as the iki.fi 
account is so spamfested.)

Now it would be also interesting to compute the positions (and values) 
where this sequence A114717
obtains A) distinct new values (i.e. 1,2,5,14,48,42,2452,462, etc. and 
the positions where they occur, i.e. 1,6,12,24,30,36,...)
and B) records (i.e. 1,2,5,14,48,2452, and their positions: 
1,6,12,24,30,60, ....)
and does any of these four sequences occur already in OEIS, and if not, 
then whether they give
us a yet another useful measure of "increasing compositeness" ?


Yours,

Antti Karttunen