Schur's Partition Theorem

Author

Eric W. Weisstein
December 22, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/SchursPartitionTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/SchursPartitionTheorem.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Partitions

a[n_]:=a[n]=Select[partitions[n],And@@Function[m,Function[c,Mod[m,6]==c]/@Or@@{1,5}]/@#&]

b[n_]:=b[n]=Select[partitions[n],Max[Length/@Split[#]]==1&&And@@Function[m,Function[c,Mod[m,3]==c]/@Or@@{1,2}]/@#&]

c[n_]:=c[n]=Module[{p=Partition[#,2,1]&/@partitions[n],cc},
cc=Append[First/@#,#[[-1,-1]]]&/@(DeleteCases[Select[p,And@@Function[f,Subtract@@f>=If[Mod[f,3]=={0,0},6,3]]/@#&],{}]);
Prepend[cc,{n}]
]

a[15]

b[15]

c[15]

Table[Length/@{a[n],b[n],c[n]},{n,10}]

Length/@Table[a[n],{n,20}]

CoefficientList[Series[QPochhammer[-q,q^3]QPochhammer[-q^2,q^3],{q,0,100}],q]

Identities

(QPochhammer2[-q, (q^3)]) (QPochhammer2[-(q^2), (q^3)])==(((QPochhammer2[(q^2), (q^6)]) (QPochhammer2[(q^4), (q^6)]))/((QPochhammer2[q, (q^3)]) (QPochhammer2[(q^2), (q^3)])))

FullSimplify[((QPochhammer[(q^2),(q^6)]QPochhammer[(q^4),(q^6)])/(QPochhammer[q,(q^3)]QPochhammer[(q^2),(q^3)]))]

FullSimplify[((QPochhammer[(q^2),(q^6)]QPochhammer[(q^4),(q^6)])/(QPochhammer[q,(q^3)]QPochhammer[(q^2),(q^3)]))-QPochhammer[-q,q^3]QPochhammer[-q^2,q^3]]

(1/((QPochhammer2[q, (q^6)]) (QPochhammer2[(q^5), (q^6)])))==(QPochhammer2[-q, (q^3)]) (QPochhammer2[-(q^2), (q^3)])

FullSimplify[(1/(QPochhammer[q,(q^6)]QPochhammer[(q^5),(q^6)]))-QPochhammer[-q,q^3]QPochhammer[-q^2,q^3]]

Subtract@@@Table[{(1/(QPochhammer[q,(q^6)]QPochhammer[(q^5),(q^6)])),QPochhammer[-q,q^3]QPochhammer[-q^2,q^3]}/.q->RandomReal[WorkingPrecision->50],{100}]//Chop//Union

((QPochhammer2[{(q^2),(q^4)}, (q^6)])/(QPochhammer2[{q,(q^2),(q^4),(q^5)}, (q^6)]))==(1/((QPochhammer2[q, (q^6)]) (QPochhammer2[(q^5), (q^6)])))

((Times@@QPochhammer[{(q^2),(q^4)},(q^6)])/(Times@@QPochhammer[{q,(q^2),(q^4),(q^5)},(q^6)]))