coverage of linear partitions

[wouter meeussen]

analogous to my earlier mail, but for 'linear' in stead of plane partitions:

the classification of partitions of n according to the number of ways it can be
'extended' to a partition of n+1 by adding 1 element to it.
Stated differently, it counts how many partitions of n have k different partitions of n+1 just
covering it :

Surprise: it equals  A060177 :

Name :      Triangle of generalized sum of divisors function, read by rows.
Comments :  Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).
References P. A. MacMahon, Divisors of numbers and their continuations in the
              theory of partitions, Proc. London Math. Soc., (2) 19 (1919), 75 - 113;
              Coll. Papers II, pp. 303 - 341.
Formula :   T(n, k) = Partitions of n using only k types of piles.


What is the similarity between "using only k types of piles" and "having k majors" ?
Do 'types of piles' refer to 'distinct elements' of a partition? with
{5,3,3,1,1,1} only having 3 types of piles? fivers, three-ers and one-ers?
That would make sense, since you could extend the 5 to 6, or the first 3 to 4, or the first 1 to 2,
or just add a 1 as new element, giving {5,3,3,1,1,1,1}.

go figure ;-))


Wouter.



btw,
I tried to make the GF in A060177 work, and got the slowest GF ever, taking minutes!
Who can come up with an improvement over:

it = Table[tem = Array[a, n];
      argu = q^(Plus @@ tem)/(Times @@ ((1-q^#) & /@ tem));
      Sum[argu,
        Evaluate[Sequence @@ Thread[{tem,Join[{1},1+Drop[tem,-1]],24}]]],
 {n, 1, 6}];
CoefficientList[Series[#, {q, 0, 24}], q] & /@ it