A156054 - OEIS

%I #4 Mar 30 2012 18:34:50

%S 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,

%T 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

%N Triangle read by rows: t(n,m)=2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m].

%C Row sums are:

%C {1, 2, 3, 6, 6, 10, 14, 18, 22, 34, 46,...}.

%C As q partitions are closely related to fermion theory, this sequence is important.

%C This sequence gives a very pretty new fractal:

%C Clear[a, b];

%C a = Table[Table[t[n, m], {m, 0, n}], {n, 0, 64}];

%C b = Table[If[m <= n, Mod[a[[n]][[m]], 2], 0], {m, 1, Length[a]}, {n, 1, Length[a]}];

%C ListDensityPlot[b, Mesh -> False, Frame -> False]

%C The statistical thermodynamics of Bosons and Fermions

%C are based on the two types of partitions

%C and each type of partition is based on products of two types of combinations at

%C large n numbers.

%C Boson combinations are:

%C Boson(n,m)=(n+m-1)!/((n-1)!*m!)

%C Fermion combinations are:

%C Fermion(n,m)=Binomial[n,m]

%C Distribution laws:

%C boson:x=(a0+b0*x(i))=n(i)

%C f(i)=n(i)/(Exp[(a0+b0*x(i)]-1)=x/(Exp[x]-1): Bernoulli numbers

%C Fermion:x=(a0+b0*x(i));n(i)=2

%C f(i)=n(i)/(Exp[(a0+b0*x(i)]-1)=2/(Exp[x]+1): Euler numbers

%C p partitions : ( Boson like): unrestricted partition

%C Sum[p[n]*x^n,{n,0,Infinity}]=Product[1/(1-x^k),{k,1,Infinity}]

%C q partitions: ( Fermion like: 2 parts): partitions into distinct parts

%C Sum[q[n]*x^n,{n,0,Infinity}]=Product[1+x^k),{k,1,Infinity}]

%C The distributions associated are Bernoulli numbers for Bosons and ( Planck as degenerate)

%C and Euler numbers for Fermions.

%F t(n,m)=2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m].

%e {1},

%e {1, 1},

%e {1, 1, 1},

%e {1, 2, 2, 1},

%e {1, 1, 2, 1, 1},

%e {1, 2, 2, 2, 2, 1},

%e {1, 2, 3, 2, 3, 2, 1},

%e {1, 2, 3, 3, 3, 3, 2, 1},

%e {1, 2, 3, 3, 4, 3, 3, 2, 1},

%e {1, 3, 4, 4, 5, 5, 4, 4, 3, 1},

%e {1, 3, 5, 5, 6, 6, 6, 5, 5, 3, 1}

%t Clear[f];

%t f[n_, m_] = 2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m];

%t Table[Table[f[n, m], {m, 0, n}], {n, 0, 10}];

%t Flatten[%]

%K nonn,tabl,uned

%O 0,8

%A _Roger L. Bagula_, Feb 02 2009

%E Spelling corrected by _Jason G. Wurtzel_, Aug 23 2010