A156054 Triangle read by rows: t(n,m)=2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m].

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

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.

Links

Table of n, a(n) for n=0..65.

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

Sequence in context: A212810 A072344 A140500 * A030616 A066422 A274888

Adjacent sequences: A156051 A156052 A156053 * A156055 A156056 A156057

Keyword

nonn,tabl,uned

Author

Roger L. Bagula, Feb 02 2009

Extensions

Spelling corrected by Jason G. Wurtzel, Aug 23 2010

Status

approved