OFFSET
0,8
COMMENTS
Row sums are:
{1, 2, 3, 6, 6, 10, 14, 18, 22, 34, 46,...}.
As q partitions are closely related to fermion theory, this sequence is important.
This sequence gives a very pretty new fractal:
Clear[a, b];
a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 64}];
b = Table[If[m <= n, Mod[a[[n]][[m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];
ListDensityPlot[b, Mesh -> False, Frame -> False]
The statistical thermodynamics of Bosons and Fermions
are based on the two types of partitions
and each type of partition is based on products of two types of combinations at
large n numbers.
Boson combinations are:
Boson(n,m)=(n+m-1)!/((n-1)!*m!)
Fermion combinations are:
Fermion(n,m)=Binomial[n,m]
Distribution laws:
boson:x=(a0+b0*x(i))=n(i)
f(i)=n(i)/(Exp[(a0+b0*x(i)]-1)=x/(Exp[x]-1): Bernoulli numbers
Fermion:x=(a0+b0*x(i));n(i)=2
f(i)=n(i)/(Exp[(a0+b0*x(i)]-1)=2/(Exp[x]+1): Euler numbers
p partitions : ( Boson like): unrestricted partition
Sum[p[n]*x^n,{n,0,Infinity}]=Product[1/(1-x^k),{k,1,Infinity}]
q partitions: ( Fermion like: 2 parts): partitions into distinct parts
Sum[q[n]*x^n,{n,0,Infinity}]=Product[1+x^k),{k,1,Infinity}]
The distributions associated are Bernoulli numbers for Bosons and ( Planck as degenerate)
and Euler numbers for Fermions.
FORMULA
t(n,m)=2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m].
EXAMPLE
{1},
{1, 1},
{1, 1, 1},
{1, 2, 2, 1},
{1, 1, 2, 1, 1},
{1, 2, 2, 2, 2, 1},
{1, 2, 3, 2, 3, 2, 1},
{1, 2, 3, 3, 3, 3, 2, 1},
{1, 2, 3, 3, 4, 3, 3, 2, 1},
{1, 3, 4, 4, 5, 5, 4, 4, 3, 1},
{1, 3, 5, 5, 6, 6, 6, 5, 5, 3, 1}
MATHEMATICA
Clear[f];
f[n_, m_] = 2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m];
Table[Table[f[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Feb 02 2009
EXTENSIONS
Spelling corrected by Jason G. Wurtzel, Aug 23 2010
STATUS
approved