[seqfan] Re: interesting idea, weird ramifications, as yet unsolved

[Wouter Meeussen]

We can look at the question asked by 'Cesare' in a wider scope:

define xx(0) as the vector {x_1, ... , x_n} and define xx(1) as the outer 
product Sum(i=1..n,j=1..n;  x_i x_j ) , subtract the diagonal elements 
Sum(i=1..n ; x_i x_i) and divide by two. This extracts only the terms below 
the diagonal of the outer product.
Repeat this procedure up to xx(k).
The result is a symmetric polynomial in n variables of degree 2^k.

The number of terms (monomials) as function of n and k is :

n=3 ; 3,  3,   3,      3, ...
n=4 ; 4,  6, 15,  102,      5010,          12 367 734, 76 069 184 710 488, 
...
n=5 ; 5, 10, 45, 975, 470 025, 110 131 246 500, ...


for n=4, there are 12 monomials containing 3 variables, all others have 4 
variables.
for n=5, I find 30 monomials in 3 variables, rest in 4 or 5 variables
(all for k>1 of course)

nice detail:
the monomials in 3 variables are simple when written in terms of the 
monomial symmetric functions:
for all k and n=2,...,7 they are m({1,1}) , m({2,1,1}), m({3,3,2}), 
m({6,5,5}), m({11,11,10}), m({22,21,21}) etc.

Wouter.
(sanity checks are always appreciated)
__________________________________
in Mathematica:
Remove[X];n=4;
X[0] = Array[Subscript[x, #] &, n];
X[k_Integer /; k > 0] :=  X[k] =
List @@ Expand[(Apply[Plus, Flatten[Outer[Times, X[k - 1], X[k - 1]]]] - 
Dot[X[k - 1], X[k - 1]])/2 ];

__________________________________

-----Original Message----- 
From: Luca Petrone
Sent: Tuesday, June 09, 2020 7:20 AM
To: Sequence Fanatics Discussion list ; Wouter Meeussen
Subject: Re: [seqfan] interesting idea, weird ramifications, as yet unsolved

Of course there are "easy" formulas to calculate how many elements of X_N 
have, for example, k repetition of one element, but things get non-sensely 
complicated of you are asking k repetions of one item, l repetitions of 
another item and so on: is really this the question? and what is it the 
meaning of it?
Best Regards,
Luca
> Il 7 giugno 2020 alle 20.16 Wouter Meeussen <wouter.meeussen at telenet.be> 
> ha scritto:
>
>
> “Pairwise combinations of distinct elements”
> see https://math.stackexchange.com/questions/3708151
>
> my feeling is that Algebraic Combinatorics should clarify this, reducing 
> it to a well known pattern.
> I tried but failed dismally.
>
> It boils down to a list of monomials X_(k+1) generated by
>
> (X_k ** X_k  - X_k . X_k )/2 where “**” represents the outer product.
>
> The author (‘Cesare’) only asks for a very limited property of these 
> multinomials, but there is a lot of other ‘enumerative combinatorics’ 
> going on here.
>
> Does anyone recognise the patten?
>
> Wouter.
>
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