Module series

class cvolume.series.PartitionFunctions

Bases: object

AC_formulae = {(2,): <function get_Fs2>, (2, 2): <function get_Fs22>, (2, 2, 2): <function get_Fs222>, (2, 3): <function get_Fs23>, (2, 4): <function get_Fs24>, (3,): <function get_Fs3>, (3, 3): <function get_Fs33>, (4,): <function get_Fs4>, (5,): <function get_Fs5>, (6,): <function get_Fs6>, (7,): <function get_Fs7>}

Z_formulae = {(2,): <function get_Zs2>}

partition_function(w)

Return the partition function F truncated at weight w.

EXAMPLE:

The partition function truncated at weight 10:

sage: from cvolume import Fs
sage: Fs.partition_function(10)
1/6*t0^3*t1^3 + 1/8*t0^4*t1*t2 + 1/120*t0^5*t3 + 1/6*t0^3*t1^2 + 1/120*t1^5 + 1/24*t0^4*t2 + 1/6*t0*t1^3*t2 + 1/6*t0^2*t1*t2^2 + 1/8*t0^2*t1^2*t3 + 7/144*t0^3*t2*t3 + 1/36*t0^3*t1*t4 + 1/576*t0^4*t5 + 1/6*t0^3*t1 + 1/96*t1^4 + 1/8*t0*t1^2*t2 + 1/24*t0^2*t2^2 + 1/16*t0^2*t1*t3 + 1/144*t0^3*t4 + 1/6*t0^3 + 1/72*t1^3 + 1/12*t0*t1*t2 + 7/1440*t2^3 + 1/48*t0^2*t3 + 29/1440*t1*t2*t3 + 29/5760*t0*t3^2 + 1/192*t1^2*t4 + 11/1440*t0*t2*t4 + 1/288*t0*t1*t5 + 1/2304*t0^2*t6 + 1/48*t1^2 + 1/24*t0*t2 + 29/5760*t2*t3 + 1/384*t1*t4 + 607/2903040*t4^2 + 1/1152*t0*t5 + 503/1451520*t3*t5 + 77/414720*t2*t6 + 5/82944*t1*t7 + 1/82944*t0*t8 + 1/24*t1 + 1/1152*t4 + 1/82944*t7

reset()

shifts = {(2,): 3, (2, 2): 6, (2, 2, 2): 9, (2, 3): 7, (2, 4): 8, (3,): 4, (3, 3): 8, (4,): 5, (5,): 6, (6,): 7, (7,): 8}

times = {(): <function <lambda>>, (2,): <function <lambda>>, (2, 2): <function <lambda>>, (2, 2, 2): <function <lambda>>, (2, 3): <function <lambda>>, (2, 4): <function <lambda>>, (3,): <function <lambda>>, (3, 3): <function <lambda>>, (4,): <function <lambda>>, (5,): <function <lambda>>, (6,): <function <lambda>>, (7,): <function <lambda>>}

cvolume.series.coeff(par)

Return the coefficient of the monomial correspoding to the partition par in the partition function F.

EXAMPLE:

Here are some examples of the coefficients of the partition function in low weight:

sage: from cvolume.series import coeff
sage: [coeff(Partition([1,1,1])),coeff(Partition([2,2,2])),coeff(Partition([4,4,1])),coeff(Partition([1,1,1,1]))]
[1, 1/12, 29/2880, 0]

cvolume.series.get_Fs2(F)

cvolume.series.get_Fs22(F)

cvolume.series.get_Fs222(F)

cvolume.series.get_Fs23(F)

cvolume.series.get_Fs24(F)

cvolume.series.get_Fs3(F)

cvolume.series.get_Fs33(F)

cvolume.series.get_Fs4(F)

cvolume.series.get_Fs5(F)

cvolume.series.get_Fs6(F)

cvolume.series.get_Fs7(F)

cvolume.series.get_Zs2(Z)

cvolume.series.get_Zs22(Z)

cvolume.series.get_Zs3(Z)

cvolume.series.monom(par)

Given a partition par = \(\left[0^{i_0},1^{i_1},\ldots,n^{i_n}\right]\) return a monomial (in Multivariate Polynomial Ring over Q) \({t_0}^{i_0}\cdot\ldots\cdot{t_n}^{i_n} \big/ {i_0}!\cdot\ldots\cdot{i_n}!\).

EXAMPLE:

Here we generate all monomials of weight 5:

sage: from cvolume.series import monom
sage: [monom(l) for l in Partitions(5)]
[t4, t0*t3, t1*t2, 1/2*t0^2*t2, 1/2*t0*t1^2, 1/6*t0^3*t1, 1/120*t0^5]