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

[Wouter Meeussen]

An other case of a near match that turns out to be misleading:
it's not  R. Zumkeller's A156145

A156145 = 1, 2, 3, 5, 8, 13, 20, 31, 46, 68, 98, 140, *195*, 271, 370, 
502, 673, 897, 1183, 1553, 2021, 2618, ...

but

A??????? = 1, 2, 3, 5, 8, 13, 20, 31, 46, 68, 98, 140, *196*, 273, 374, 
509, 685, 916, 1213, 1598, 2088, 2715, ...

simply the reverse-lexicographic ranks of the partitions of 2n into at 
most 2 parts.
Part of the mystery solved, but the magic link with Hanna's GF remains 
puzzling.

(checked up to 2n=24)

Wouter.

ps. should the trivial A?????? be submitted?

---------------------------------------------------------------

Op 9/03/2023 om 21:51 schreef 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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