[seqfan] Hanna's table A322210 expanded and condensed again gives ... R. Zumkeller's A156145, duh? who ordered that?

[Wouter Meeussen]

how weird can it get?

A156145 = 1, 2, 3, 5, 8, 13, 20, 31, 46, 68, 98, ...
Number of partitions of 9*n-8 into parts having in decimal 
representation digital root 1.
alias A010888(9*n-8) = 1;
but then, that A010888 = Digital root of n (repeatedly add the digits of 
n until a single digit is reached).

Is this numerology or Number Theory? I'll show the connection below.


And now for something (apparently) completely different:

Hanna's A322210 is the 2-dim coefficientlist of x,y in
prod(n=0,N; 1/(1 - ( x^n +  y^n) ) )

An old trick to decompose the entries in the table is to parametrise like
prod(n=0,N;1/(1 - (l[n] x^n + l[n] y^n)) )
where I choose the dummy variable l ('el') for its typographical 
compactness.
The 'expanded' table then starts like:

1                       l(1) l(1)^2+l(2)      ...
l(1)                2 l(1)^2                    3 l(1)^3+l(2) l(1)
l(1)^2+l(2)    3 l(1)^3+l(2) l(1)    6 l(1)^4+2 l(2) l(1)^2+2 l(2)^2
...
remark that setting all l(..) equal to 1 reverts to A322210

each entry above can be read as a polynomial of partition valued functions :
f{}                   f{1}                       f{1,1}+f{2}
f{1}               2f{1,1}                  3f{1,1,1}+f{2,1}
f{1,1}+f{2}   3f{1,1,1}+f{2,1}  6f{1,1,1,1}+2f{2,1,1}+2f{2,2}
...

thes expressions get very unwieldly, but if we choose the Schur function 
in the role of 'f', then this verbose expanded table condenses again 
nicely to:

1              chi(1,1) chi(2,1)                               chi(3,1) ...
chi(1,1)  chi(2,1)+chi(2,2)   chi(3,1)+chi(3,2) chi(4,1)+chi(4,2)  ....
chi(2,1)  chi(3,1)+chi(3,2)   chi(4,1)+chi(4,2)+chi(4,3) 
chi(5,1)+chi(5,2)+chi(5,3) ...
...               ... ... chi(6,1)+chi(6,2)+chi(6,3)+chi(6,5)  ...

or
*table( i=1..n,j=1..n ; sum(chi(i, A156145( min(i,j) ) ) )*

and chi( a,b ) is the list of characters in row b of the symmetric group 
of size a.
(Could be called 'decomposition into orthonormal Schur components' or 
sumptin like that)

Example:
chi(4,1)+chi(4,2)+chi(4,3) equals
{1, 1, 1, 1, 1}+{-1, 0, -1, 1, 3}+{0, -1, 2, 0, 2} = {0, 0, 2, 2, 6}
0*f{4}+0*f{3,1}+2*f{2,2}+2*f{2,1,1}+6*f{1,1,1,1}
as in row 3 column 3 of the 'polynomial of partition valued functions'.

Who in his right mind expected such simple decomposition in terms of 
characters,
and who saw the relevance of A156145 coming from afar? I didn't.

All the above was checked up to T(10,10).

Wouter.

--------------------------------------------------------------------------
pro memori :  WARNING
Mathematica code beyond, (slippery even when dry)

w = 10;
it1 =  Expand at CoefficientList[
    Series[Product[1/(1 - (l[n] x^n + l[n] y^n)), {n, w}], {x, 0,
      w}, {y, 0, w}], {x, y}];

it2 = it1 //. l[u_]^(k_) :> l[Sequence @@ Table[u, k]] //.
      l[u__] l[v__] :> l[u, v] /. l[li__] :> s at Reverse[Sort[{li}]] /.
    s[pa__] :> s^rankpartition[pa];

it3 = Map[Rest at CoefficientList[#, s] &, it2, {2}];

it4 = MapIndexed[q[#1, Tr[#2] - 2] &, it3, {2}];

it5 = it4 /.
   q[u : {__},
     i_] :> ( Map[chars, Partitions[i]]. (cycleclasses[i]/i! u)).
     Array[Subscript[\[Chi], i], PartitionsP at i]

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