cp's OEIS Frontend

Call C(p, [alpha], g) the number of partitions of the cyclically ordered set [p], of cyclic type [alpha], and of genus g (genus g Faa di Bruno coefficients of type [alpha]). The number C(3n, [3^n], g) of genus g partitions of the set [3n] into n blocks of length 3 is given by A001764 when g=0 (non-crossing partitions), by [Zuber] when g = 1 (see A371250) and g=2 (this sequence). One has C(n=6,[3^2],2) = 1, C(n=9, [3^3], 2) = 144, etc.

Call C(p,[alpha],g) the number of partitions of the cyclically ordered set [p], of cyclic type [alpha], and of genus g (genus g Faa di Bruno coefficients of type [alpha]). The number C(3n,[3^n],g) of genus g partitions of the set [3n] into n blocks of length 3 is given by A001764 if g=0 (non-crossing partitions), by [Zuber] when g=1 (this sequence) and g=2, see A371251. One has C(n=6,[3^2],1) = 6, C(n=9,[3^3],1) = 102, etc.

Genus-dependent Stirling numbers of the second kind S2(n,k,g), 1 <= n, 1 <= k <= n, 0 <= g <= floor((n-1)/2). This is an infinite three-dimensional array. Its first 15 rows (n:1..15) are given by the table (see Links) taken from the article by Robert Coquereaux and Jean-Bernard Zuber (where a transpose of this table is given), see p. 32. These 15 rows determine 589 entries of the sequence (Data).

Example: the numbers S2(5,k,0), k=1..5, are {1,10,20,10,1} and appear on line 5, column 1; the numbers S2(5,k,1), k=1..5, are {0,5,5,0,0} and appear on line 5, column 2. Values of S2(n,k,g) for g > floor((n-1)/2) are equal to 0 and are not displayed.

Summing S2(n,k,g) over k gives genus-dependent Bell numbers B(n,g), A370235. Summing S2(n,k,g) over g gives S2(n,k), the Stirling numbers of the second kind A008277. Summing S2(n,k,g) over k and g gives the Bell numbers B(n), A000110. Example: S2(5,k,0) = 1, 10, 20, 10, 1 and S2(5,k,1) = 0, 5, 5, 0, 0 for k = 1..5; therefore S2(5,k) = 1, 15, 25, 10, 1, B(5,0) = 42, B(5,1) = 10, and B(5) = 52.

Call B(n, g) the number of genus g partitions of a set with n elements (genus-dependent Bell number). Then a(n) = B(n, 3) with B(8, 3) = 1.

a(8) = 1 through a(15) = 565256120 were explicitly determined by listing of partitions of an n-set and selecting those of genus 3.

The coefficients of the sixth-degree polynomial appearing in the numerator of the conjectured formula were determined by using experimental values for a(8) up to a(14); the term a(15) given by the formula agrees with the experimental value.

Using the conjectured formula for a(n) gives the following terms for n=16..20 : 4593034160, 35025118700, 253374008888, 1753071498620, 11675101781850. The E.g.f. given in the Formula section is obtained from the conjectured formula for a(n).

The formula given below was conjectured by Martha Yip and proved by Robert Cori and Gábor Hetyei.

More generally one may consider genus-dependent Stirling numbers S(n, k, g) that count the partitions of genus g and k parts of the n-set.

Then T(n, k) = S(n, k, 1). See Robert Coquereaux and Jean-Bernard Zuber.

This lattice is associated with the exceptional module-category E_21(SU(3)) over the fusion (monoidal) category A_21(SU(3)).

The Grothendieck group of the former, a finite abelian category, is a Z+ - module over the Grothendieck ring of the latter, with a basis given by isomorphism classes of simple objects.

Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.

The classification of module-categories over A_k(SU(3)) was done, using another terminology, by P. Di Francesco and J.-B Zuber, and by A. Ocneanu (see refs below): it contains several infinite families that exist for all values of the positive integer k (among others one finds the A_k(SU(3)) themselves and the orbifold series D_k(SU(3))), and several exceptional cases for special values of k.

To every such module-category one can associate a set of hyper-roots (see refs below) and consider the corresponding lattice, denoted by the same symbol.

E_k(SU(3)), with k=21, is one of the exceptional cases; other exceptional cases exist for k=5 and k=9. It is also special because it has self-fusion (it is flat, in operator algebra parlance).

E_21(SU(3)) has r=24 simple objects. The rank of the lattice is 2r=48. Det =3^12. This lattice, using k=21, is defined by 2r(k+3)^2/3=9216 hyper-roots of norm 6.

The first shell is made of vectors of norm 4, they are not hyper-roots, and the second shell, of norm 6, contains not only the hyper-roots but other vectors as well. Note: for lattices of type A_k(SU(3)), vectors of shortest length and hyper-roots coincide, here this is not so.

The lattice is rescaled (q --> q^2): its theta function starts as 1 +144*q^4 + 64512*q^6 +... See example.

This theta series is an element of Gamma_0(3) of weight 24 and dimension 9. - Andy Huchala, May 14 2023

This lattice is associated with the exceptional module-category E_9(SU(3)) over the fusion (monoidal) category A_9(SU(3)).

The Grothendieck group of the former, a finite abelian category, is a Z+ - module over the Grothendieck ring of the latter, with a basis given by isomorphism classes of simple objects.

Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.

The classification of module-categories over A_k(SU(3)) was done, using another terminology, by P. Di Francesco and J.-B Zuber, and by A. Ocneanu (see refs below): it contains several infinite families that exist for all values of the positive integer k (among others one finds the A_k(SU(3)) themselves and the orbifold series D_k(SU(3))), and several exceptional cases for special values of k.

To every such module-category one can associate a set of hyper-roots (see refs below) and consider the corresponding lattice, denoted by the same symbol.

E_k(SU(3)), with k=9, is one of the exceptional cases; other exceptional cases exist for k=5 and k=21. It is also special because it has self-fusion (it is flat, in operator algebra parlance).

E_9(SU(3)) has r=12 simple objects. The rank of the lattice is 2r=24. Det =2^24. This lattice, using k=9, is defined by 2*r*(k+3)^2/3=1152 hyper-roots of norm 6. The first shell is made of vectors of norm 4, they are not hyper-roots, and the second shell, of norm 6, contains not only the hyper-roots, but other vectors as well. Note: for lattices of type A_k(SU(3)), vectors of shortest length and hyper-roots coincide, here this is not so.

The lattice is rescaled (q --> q^2): its theta function starts as 1 + 756*q^4 + 5760*q^6 +... See example.

This theta series is an element of the space of modular forms on Gamma_0(8) of weight 12 and dimension 13. - Andy Huchala, May 14 2023

This lattice is associated with the exceptional module-category E_5(SU(3)) over the fusion (monoidal) category A_5(SU(3)).

The Grothendieck group of the former, a finite abelian category, is a Z+ - module over the Grothendieck ring of the latter, with a basis given by isomorphism classes of simple objects.

Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.

The classification of module-categories over A_k(SU(3)) was done, using another terminology, by P. Di Francesco and J.-B Zuber, and by A. Ocneanu (see refs below): it contains several infinite families that exist for all values of the positive integer k (among others one finds the A_k(SU(3)) themselves and the orbifold series D_k(SU(3))), and several exceptional cases for special values of k.

To every such module-category one can associate a set of hyper-roots (see refs below) and consider the corresponding lattice, denoted by the same symbol.

E_k(SU(3)), with k=5, is one of the exceptional cases; other exceptional cases exist for k=9 and k=21. It is also special because it has self-fusion (it is flat, in operator algebra parlance).

E_5(SU(3)) has r=12 simple objects. The rank of the lattice is 2r=24. Det =2^30. This lattice, with k=5, is defined by 2 * r * (k+3)^2/3=512 hyper-roots of norm 6. They are also the vectors of shortest length (so, vectors of shortest length and hyper-roots coincide, like for lattices of type A_k(SU(3))). Minimal norm is 6.

The lattice is rescaled (q --> q^2): its theta function starts as 1 + 512*q^6 + 11232*q^8 +... See example.

This theta series is an element of the space of modular forms on Gamma_0(16) of weight 12 and dimension 25. - Andy Huchala, May 14 2023

This lattice is the k=6 member of the family of lattices of SU(3) hyper-roots associated with the module-category D_k(SU(3)) over the fusion (monoidal) category A_k(SU(3)).

The Grothendieck group of the former, a finite abelian category, is a Z+ - module over the Grothendieck ring of the latter, with a basis given by isomorphism classes of simple objects.

Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.

The classification of module-categories over A_k(SU(3)) was done, using another terminology, by P. Di Francesco and J.-B Zuber, and by A. Ocneanu (see refs below): it contains several infinite families that exist for all values of the positive integer k (among others one finds the A_k(SU(3)) themselves and the orbifold series D_k(SU(3))), and several exceptional cases for special values of k.

To every such module-category one can associate a set of hyper-roots (see refs below) and consider the corresponding lattice, denoted by the same symbol.

Members of the subfamily D_{3s} are special because they have self-fusion (they are flat, in operator algebra parlance). D_6(SU(3)) is the second smallest member of the D_{3s} family (s=2).

With k=6 there are r = ((k+1)*(k+2)/2 - 1)/3 + 3 = 12 simple objects. The rank of the lattice is 2r=24. The lattice is defined by 2*r*(k+3)^2/3 = 648 hyper-roots of norm 6. Det = 3^18. The first shell is made of vectors of norm 4, they are not hyper-roots, and the second shell, of norm 6, contains not only the hyper-roots but other vectors. Note: for lattices of type A_k(SU(3)), vectors of shortest length and hyper-roots coincide, here this is not so.

The lattice is rescaled (q --> q^2): its theta function starts as 1 + 162*q^4 + 2322*q^6 + ... See example.

This theta series is an element of the space of modular forms on Gamma_0(27) of weight 12 and dimension 36. - Andy Huchala, May 14 2023

This lattice is the k=3 member of the family of lattices of SU(3) hyper-roots associated with the module-category D_k(SU(3)) over the fusion (monoidal) category A_k(SU(3)).The Grothendieck group of the former, a finite abelian category, is a Z+ - module over the Grothendieck ring of the latter, with a basis given by isomorphism classes of simple objects.

Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.

The classification of module-categories over A_k(SU(3)) was done, using another terminology, by P. Di Francesco and J.-B Zuber, and by A. Ocneanu (see refs below): it contains several infinite families that exist for all values of the positive integer k (among others one finds the A_k(SU(3)) themselves and the orbifold series D_k(SU(3))), and several exceptional cases for special values of k.

To every such module-category one can associate a set of hyper-roots (see refs below) and consider the corresponding lattice, denoted by the same symbol.

Members of the sub-family D_{3s} are special because they have self-fusion (they are flat, in operator algebra parlance). D_3(SU(3)) is the smallest member of the D_{3s} family (s=1).

With k=3 there are r=((k+1)(k+2)/2 -1)/3+3=6 simple objects. The rank of the lattice is 2r=12. The lattice is defined by 2r(k+3)^2/3=144 hyper-roots of norm 6. Det =3^12. The first shell is made of vectors of norm 4, they are not hyper-roots, and the only vectors of the lattice that belong to the second shell, of norm 6, are precisely the hyper-roots. Note: for lattices of type A_k(SU(3)), vectors of shortest length and hyper-roots coincide, here this is not so.

The lattice is rescaled (q --> q^2): its theta function starts as 1 + 36*q^4 + 144*q^6 +... See example.