Hanna's triangle A096651

[Paul D. Hanna]

     Regarding the comment:
> not EIS-able yet, since the mere existence of the polynomials is
> pure conjecture itself.
this has been addressed by George Andrews in his book (which I bought), 
"The Theory of Partitions", section: 11.4 Higher-Dimensional Partitions. 

There, on pages 189-197 he establishes the fact that the polynomials you
mentioned do indeed exist, and gives a the Binomial coefficients needed
to calculate n-dimensional partitions up to 6 in terms of n. 
  
These Binomial coefficients that generate n-dimensional partitions are
recorded in OEIS at:
   http://www.research.att.com/projects/OEIS?Anum=A096806
Further, the inverse binomial transform of the diagonals of this triangle
A096806 (also recorded in A096806) seem to provide some hints at a
pattern for further rows.
 
      Why is it that the row sums of A096651^n form the n-dimensional
partitions? 
First note that triangle A096651 forms the coefficients of the
multidimensional partition transform of n-dimensional partitions into
(n+1)-dimensional partitions.  
The property that the row sums of matrix power A096651^n form the
n-dimensional partitions is due to the fact that the zero-dimensional
partitions are defined as 1 for all integers.  
Applying the multidimensional partition transform (A096651) n-times upon
the sequence of 1's thus generates the n-dimensional partitions, which in
turn also equal the row sums of the n-th matrix power of A096651.    
 
Wouter, in my analysis I also observed the possible connection to A000806
- very interesting if so. 
One minor correction: I think that for
> T(n+6,n)=(2400-2292*n-330*n^2+180*n^3+210*n^4+72*n^5)/120
you meant:
   T(n+6,n)= (1200-1146*n-165*n^2+90*n^3+105*n^4+36*n^5)/120
which has the 36 in the high order term.
 
Thanks,
      Paul
 
 
On Sat, 23 Oct 2004 23:30:34 +0200 "wouter meeussen"
<wouter.meeussen at pandora.be> writes:
> for those who remember, a lower triangular matrix H such that the row 
> sums of H^k give the count of
> k-dimensional partitions. Not the end of math as we know it, but 
> still ... (hmm).
> 
> News:
> the few regularities of H[n,m] known are:
> T(n,n)=1
> T(n+1,n)=1
> T(n+2,n)=n
> T(n+3,n)=1
> T(n+4,n)=(0+5*n+0*n^2+n^3)/6
> T(n+5,n)=(-48+90*n-7*n^2-6*n^3-5*n^4)/24
> T(n+6,n)=(2400-2292*n-330*n^2+180*n^3+210*n^4+72*n^5)/120
> 
> with sketchy hints towards:
>
T(n+7,n)=(-16560+3600*n+17554*n^2+1395*n^3-3185*n^4-1755*n^5-329*n^6)/720
> 
> now it looks like the coefficients of the highest powers in n 
> follow:
> A000806 = 1,0,1,-5,36,-329,3655,-47844,721315,-12310199,234615096
> aka
>  Name:      Bessel polynomial y_n(-1).
>  References :
>  G. Kreweras and Y. Poupard, Sur les partitions en paires d'un 
> ensemble
>  fini totalement ordonne, Publications de l'Institut de Statistique
>  de l'Universite de Paris, 23 (1978), 57-74.
> 
> Does this ring a bell with anyone?
> 
> 
> Can anyone extend the 5-, 6- and 7-dimensional partitions beyond the 
> current
> A000390, A000416 and A000427 ?
> that could definitively disprove this conjecture.
> And disproving conjectures is, .., well, like *fun*
> 
> btw, the coefficients of the polynomials so far:
> 1
> 0, 1
> 2, 0, 0
> 0, 5, 0, 1
> -48, 90, -7, -6, -5
> 1200, -1146, -165, 90, 105, 36
> -16560, 3600, 17554, 1395, -3185, -1755, -329
> 
> not EIS-able yet, since the mere existence of the polynomials is
> pure conjecture itself.
> 
> W.
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