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<DIV> Regarding the comment:</DIV>
<DIV>> not EIS-able yet, since the mere existence of the polynomials
is<BR>> pure conjecture itself.</DIV>
<DIV>this has been addressed by George Andrews in his book (which I
bought), </DIV>
<DIV>"The Theory of Partitions", section: 11.4 Higher-Dimensional
Partitions. </DIV>
<DIV>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. </DIV>
<DIV> </DIV>
<DIV>These Binomial coefficients that generate n-dimensional
partitions are recorded in OEIS at:</DIV>
<DIV> <A
href="http://www.research.att.com/projects/OEIS?Anum=A096806">http://www.research.att.com/projects/OEIS?Anum=A096806</A><BR>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.</DIV>
<DIV> </DIV>
<DIV>
<DIV> Why is it that the row sums of
A096651^n form the n-dimensional partitions? </DIV>
<DIV>First note that triangle A096651 forms the coefficients of
the multidimensional partition transform of n-dimensional partitions
into (n+1)-dimensional partitions. </DIV>
<DIV>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. </DIV>
<DIV>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. </DIV></DIV>
<DIV> </DIV>
<DIV>Wouter, in my analysis I also observed the possible connection to
A000806 - very interesting if so. </DIV>
<DIV>One minor correction: I think that for</DIV>
<DIV>> T(n+6,n)=(2400-2292*n-330*n^2+180*n^3+210*n^4+72*n^5)/120</DIV>
<DIV>you meant:</DIV>
<DIV> T(n+6,n)=
(1200-1146*n-165*n^2+90*n^3+105*n^4+36*n^5)/120</DIV>
<DIV>which has the 36 in the high order term.</DIV>
<DIV> </DIV>
<DIV>Thanks,</DIV>
<DIV> Paul</DIV>
<DIV> </DIV>
<DIV> </DIV>
<DIV>On Sat, 23 Oct 2004 23:30:34 +0200 "wouter meeussen" <<A
href="mailto:wouter.meeussen@pandora.be">wouter.meeussen@pandora.be</A>>
writes:<BR>> for those who remember, a lower triangular matrix H such that
the row <BR>> sums of H^k give the count of<BR>> k-dimensional partitions.
Not the end of math as we know it, but <BR>> still ... (hmm).<BR>>
<BR>> News:<BR>> the few regularities of H[n,m] known are:<BR>>
T(n,n)=1<BR>> T(n+1,n)=1<BR>> T(n+2,n)=n<BR>> T(n+3,n)=1<BR>>
T(n+4,n)=(0+5*n+0*n^2+n^3)/6<BR>>
T(n+5,n)=(-48+90*n-7*n^2-6*n^3-5*n^4)/24<BR>>
T(n+6,n)=(2400-2292*n-330*n^2+180*n^3+210*n^4+72*n^5)/120<BR>> <BR>> with
sketchy hints towards:<BR>>
T(n+7,n)=(-16560+3600*n+17554*n^2+1395*n^3-3185*n^4-1755*n^5-329*n^6)/720<BR>>
<BR>> now it looks like the coefficients of the highest powers in n <BR>>
follow:<BR>> A000806 =
1,0,1,-5,36,-329,3655,-47844,721315,-12310199,234615096<BR>>
aka<BR>> Name: Bessel polynomial
y_n(-1).<BR>> References :<BR>> G. Kreweras and Y. Poupard,
Sur les partitions en paires d'un <BR>> ensemble<BR>> fini
totalement ordonne, Publications de l'Institut de Statistique<BR>> de
l'Universite de Paris, 23 (1978), 57-74.<BR>> <BR>> Does this ring a bell
with anyone?<BR>> <BR>> <BR>> Can anyone extend the 5-, 6- and
7-dimensional partitions beyond the <BR>> current<BR>> A000390, A000416
and A000427 ?<BR>> that could definitively disprove this conjecture.<BR>>
And disproving conjectures is, .., well, like *fun*<BR>> <BR>> btw, the
coefficients of the polynomials so far:<BR>> 1<BR>> 0, 1<BR>> 2, 0,
0<BR>> 0, 5, 0, 1<BR>> -48, 90, -7, -6, -5<BR>> 1200, -1146, -165, 90,
105, 36<BR>> -16560, 3600, 17554, 1395, -3185, -1755, -329<BR>> <BR>>
not EIS-able yet, since the mere existence of the polynomials is<BR>> pure
conjecture itself.<BR>> <BR>> W.</DIV></BODY></HTML>