Rauzy class cardinality

Cardinality of Rauzy classes

The following functions implement algorithms relative to article [Del13] and [Boi13] where are given formulas for the cardinality of Rauzy classes of permutations.

• c: number of standard labeled permutations

• d: spin difference of standard labeled permutations

• gamma_std: number of standard permutations (with given profile and marking)

• gamma_irr: number of irreducible permutations (with given profile and marking)

• delta_std: spin difference for standard permutations (with given profile and marking)

• delta_irr: spin difference for irreducible permutations (with given profile and marking)

AUTHOR:

Vincent Delecroix

surface_dynamics.interval_exchanges.rauzy_class_cardinality.bidecompositions(p)

Iterator through the pair of partitions (q1,q2) such that the union of the parts of q1 and q2 equal p.

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc
sage: list(rcc.bidecompositions(Partition([3,1])))
[([], [3, 1]), ([3], [1]), ([1], [3]), ([3, 1], [])]
sage: list(rcc.bidecompositions(Partition([2,1,1])))
[([], [2, 1, 1]),
 ([2], [1, 1]),
 ([1], [2, 1]),
 ([2, 1], [1]),
 ([1, 1], [2]),
 ([2, 1, 1], [])]

surface_dynamics.interval_exchanges.rauzy_class_cardinality.c(p)

Number of labeled standard permutations with given profile

There is an explicit formula for this number

\[c(p) = \frac{2 (n-1)!}{n+1} \left( \sum_{q \subset (p_2,p_3,\ldots,p_k)} (-1)^{s(q)-l(q)} \binom{n}{s(q)}^{-1} \right).\]

Though, for huge partition p this is not very useful. This function implements an induction formula to compute c(p).

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc

Partition of length 1:

sage: n = 7
sage: rcc.c([n]) == 2 * factorial(n-1) / (n+1)
True
sage: all(rcc.c([n]) == 2 * factorial(n-1) / (n+1) for n in range(11,18,2))
True

Partitions of length 2 with two odd numbers:

sage: p = [5,3]
sage: n = sum(p)
sage: b = binomial(n,p[0])
sage: rcc.c(p) == 2 * factorial(n-1) * (1 + 1 / binomial(n,p[0])) / (n+1)
True

sage: p = [13,5]
sage: n = sum(p)
sage: b = binomial(n,p[0])
sage: rcc.c(p) == 2 * factorial(n-1) * (1 + 1 / binomial(n,p[0])) / (n+1)
True

Partitions of length 2 with even numbers:

sage: p = [4,4]
sage: n = sum(p)
sage: b = binomial(n,p[0])
sage: rcc.c(p) == 2 * factorial(n-1) * (1 - 1 / binomial(n,p[0])) / (n+1)
True

sage: p = [10,2]
sage: n = sum(p)
sage: b = binomial(n,p[0])
sage: rcc.c(p) == 2 * factorial(n-1) * (1 - 1 / binomial(n,p[0])) / (n+1)
True

Add marked points to an integer partition:

sage: p = [3,2,2]
sage: n = sum(p)
sage: all(rcc.c(p + [1]*k) == factorial(n+k-1) / factorial(n-1) * rcc.c(p) for k in range(1,6))
True

surface_dynamics.interval_exchanges.rauzy_class_cardinality.check_marking(p, marking)

Tiny internal function that checks that marking is compatible with p.

OUTPUT:

A 3-tuple.

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc
sage: p = Partition([3,2,2])
sage: rcc.check_marking(p, (1,3,1))
(1, 3, 1)
sage: rcc.check_marking(p, (1,3,2))
(1, 3, 2)
sage: rcc.check_marking(p, (1,3,3))
Traceback (most recent call last):
...
ValueError: marking[2] is not good
sage: rcc.check_marking(p, (1,3,-1))
Traceback (most recent call last):
...
ValueError: marking[2] is not good
sage: rcc.check_marking(p, (2,3,2))
(2, 3, 2)
sage: rcc.check_marking(p, (2,3,3))
Traceback (most recent call last):
...
ValueError: wrong marking type 2

surface_dynamics.interval_exchanges.rauzy_class_cardinality.check_std_marking(p, marking)

Tiny internal function that checks the validity of marking on the partition p.

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc
sage: p = Partition([3,2,2])
sage: rcc.check_std_marking(p, (1,3,1))
(1, 3, 1)
sage: rcc.check_std_marking(p, (1,3,0))
Traceback (most recent call last):
...
ValueError: marking[2] is not good

surface_dynamics.interval_exchanges.rauzy_class_cardinality.collapse(p, i, j)

Collapses the i-th term and the j-th term of a permutation

INPUT:

• p - a partition

• i,j - two different indices of p

OUTPUT:

• a partition

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc
sage: p = Partition([4,2,1])
sage: rcc.collapse(p, 0, 1)
[5, 1]
sage: rcc.collapse(p, 0, 2)
[4, 2]
sage: rcc.collapse(p, 1, 2)
[4, 2]

surface_dynamics.interval_exchanges.rauzy_class_cardinality.d(p)

Difference between the number of odd spin parity and even spin parity standard labeled permutations with given profile

There is an explicit formula

\[d(p) = \frac{(n-1)!}{2^{(n-k)/2}}\]

where n is the sum of the partition p and k is its length.

EXAMPLES:

   sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc
   sage: p = [3,3,1]
   sage: rcc.d([3,3,1]) == factorial(6) / 2**2
   True
   sage: rcc.d([13]) == factorial(12) / 2**6
   True

Adding marked points::

   sage: p = [5,3,3]
   sage: n = sum(p)
   sage: all(rcc.d(p + [1]*k) == factorial(n+k-1) / factorial(n-1) * rcc.d(p) for k in range(1,6))
   True

surface_dynamics.interval_exchanges.rauzy_class_cardinality.delta_irr(profile, marking=None)

Spin difference for the given profile and marking

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc

The non connected strata in genus 3:

sage: from surface_dynamics import Stratum

sage: c_odd = Stratum([2,2], k=1).odd_component()
sage: c_hyp = Stratum([2,2], k=1).hyperelliptic_component()
sage: c_odd.rauzy_diagram()
Rauzy diagram with 294 permutations
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 63 permutations
sage: rcc.delta_irr([3,3]) == 294 - 63
True

sage: c_odd = Stratum([4], k=1).odd_component()
sage: c_hyp = Stratum([4], k=1).hyperelliptic_component()
sage: c_odd.rauzy_diagram()
Rauzy diagram with 134 permutations
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 31 permutations
sage: rcc.delta_irr([5]) == 134 - 31
True

A non connected strata in genus 4:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rdc
sage: a = Stratum([6], k=1)
sage: c_hyp = a.hyperelliptic_component()
sage: c_odd = a.odd_component()
sage: c_even = a.even_component()
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 127 permutations
sage: c_hyp.rauzy_class_cardinality()
127
sage: c_odd.rauzy_diagram()
Rauzy diagram with 5209 permutations
sage: c_odd.rauzy_class_cardinality()
5209
sage: c_even = a.even_component()
sage: c_even.rauzy_diagram()
Rauzy diagram with 2327 permutations
sage: c_even.rauzy_class_cardinality()
2327
sage: 5209 - 2327 - 127
2755
sage: rdc.delta_irr([7])
2755

An example with a very big Rauzy class:

sage: c = Stratum([6,6], k=1).odd_component()
sage: c.rauzy_class_cardinality()
11609364656

surface_dynamics.interval_exchanges.rauzy_class_cardinality.delta_std(profile, marking=None)

Return the difference odd-even in the given stratum

INPUT:

• p - partition with odd terms

• marking - a 3-tuple (1, n1, a) or (2, n1, n2)

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc

A ValueError is raised if the partition does not fulfill the requirement:

sage: rcc.delta_std([5,2])
Traceback (most recent call last):
...
ValueError: the profile (=[5, 2]) must contain only odd numbers

Non connected strata in genus 3 has two connected components distinguished by their spin parity:

sage: from surface_dynamics import Stratum
sage: cc_odd = Stratum([2,2], k=1).odd_component()
sage: cc_hyp = Stratum([2,2], k=1).hyperelliptic_component()
sage: d_odd = cc_odd.rauzy_diagram()
sage: d_hyp = cc_hyp.rauzy_diagram()
sage: d_odd
Rauzy diagram with 294 permutations
sage: d_hyp
Rauzy diagram with 63 permutations
sage: n_odd = sum(1 for _ in filter(lambda x: x.is_standard(), d_odd))
sage: n_hyp = sum(1 for _ in filter(lambda x: x.is_standard(), d_hyp))
sage: n_odd - n_hyp == rcc.delta_std([3,3])
True

sage: cc_odd = Stratum([4], k=1).odd_component()
sage: cc_hyp = Stratum([4], k=1).hyperelliptic_component()
sage: d_odd = cc_odd.rauzy_diagram()
sage: d_hyp = cc_hyp.rauzy_diagram()
sage: d_odd
Rauzy diagram with 134 permutations
sage: d_hyp
Rauzy diagram with 31 permutations
sage: n_odd = sum(1 for _ in filter(lambda x: x.is_standard(), d_odd))
sage: n_hyp = sum(1 for _ in filter(lambda x: x.is_standard(), d_hyp))
sage: n_odd - n_hyp == rcc.delta_std([5])
True

surface_dynamics.interval_exchanges.rauzy_class_cardinality.gamma_irr(profile=None, marking=None)

Number of permutations for the given profile and marking

INPUT:

• profile - an integer partition such that its sum plus its length is congruent to 0 modulo 2

• markings - None, an element of the profile or a 3-tuple

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc

The connected strata in genus 3:

sage: from surface_dynamics import Stratum

sage: c = Stratum([1,1,1,1], k=1).unique_component()
sage: c.rauzy_diagram()
Rauzy diagram with 1255 permutations
sage: rcc.gamma_irr([2,2,2,2])
1255

sage: c = Stratum([2,1,1], k=1).unique_component()
sage: c.rauzy_diagram()
Rauzy diagram with 2177 permutations
sage: rcc.gamma_irr([3,2,2])
2177

sage: c = Stratum([3,1], k=1).unique_component()
sage: c.rauzy_diagram()
Rauzy diagram with 770 permutations
sage: rcc.gamma_irr([4,2])
770

The non connected strata in genus 3:

sage: c_odd = Stratum([2,2], k=1).odd_component()
sage: c_hyp = Stratum([2,2], k=1).hyperelliptic_component()
sage: c_odd.rauzy_diagram()
Rauzy diagram with 294 permutations
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 63 permutations
sage: rcc.gamma_irr([3,3]) == 294 + 63
True

sage: c_odd = Stratum([4], k=1).odd_component()
sage: c_hyp = Stratum([4], k=1).hyperelliptic_component()
sage: c_odd.rauzy_diagram()
Rauzy diagram with 134 permutations
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 31 permutations
sage: rcc.gamma_irr([5]) == 134 + 31
True

surface_dynamics.interval_exchanges.rauzy_class_cardinality.gamma_std(profile, marking=None)

Return the number of standard permutations of given profile

INPUT:

• profile - an integer partition such that the its sum plus its length is congruent to 0 modulo 2

• marking - either None, an element of the profile or a 3-tuple (1, n1, a) or (2, n1, n2)

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc

A ValueError is raised if the partition does not satisfy the requirement:

sage: rcc.gamma_std([5,2])
Traceback (most recent call last):
...
ValueError: the sum of the profile (=[5, 2]) plus its length must be congruent to 0 modulo 2

The Rauzy classes associated to connected strata in genus 3:

sage: from surface_dynamics import Stratum
sage: cc = Stratum([1,1,1,1], k=1).unique_component()
sage: d = cc.rauzy_diagram()
sage: d
Rauzy diagram with 1255 permutations
sage: sum(1 for _ in filter(lambda x: x.is_standard(), d)) == rcc.gamma_std([2,2,2,2])
True

sage: cc = Stratum([2,1,1], k=1).unique_component()
sage: d = cc.rauzy_diagram()
sage: d
Rauzy diagram with 2177 permutations
sage: sum(1 for _ in filter(lambda x: x.is_standard(), d)) == rcc.gamma_std([3,2,2])
True

sage: cc = Stratum([3,1], k=1).unique_component()
sage: d = cc.rauzy_diagram()
sage: d
Rauzy diagram with 770 permutations
sage: sum(1 for _ in filter(lambda x: x.is_standard(), d)) == rcc.gamma_std([4,2])
True

The non connected strata in genus 3:

sage: cc_odd = Stratum([2,2], k=1).odd_component()
sage: cc_hyp = Stratum([2,2], k=1).hyperelliptic_component()
sage: d_odd = cc_odd.rauzy_diagram()
sage: d_hyp = cc_hyp.rauzy_diagram()
sage: d_odd
Rauzy diagram with 294 permutations
sage: d_hyp
Rauzy diagram with 63 permutations
sage: n_odd = sum(1 for _ in filter(lambda x: x.is_standard(), d_odd))
sage: n_hyp = sum(1 for _ in filter(lambda x: x.is_standard(), d_hyp))
sage: n_odd + n_hyp == rcc.gamma_std([3,3])
True

sage: cc_odd = Stratum([4], k=1).odd_component()
sage: cc_hyp = Stratum([4], k=1).hyperelliptic_component()
sage: d_odd = cc_odd.rauzy_diagram()
sage: d_hyp = cc_hyp.rauzy_diagram()
sage: d_odd
Rauzy diagram with 134 permutations
sage: d_hyp
Rauzy diagram with 31 permutations
sage: n_odd = sum(1 for _ in filter(lambda x: x.is_standard(), d_odd))
sage: n_hyp = sum(1 for _ in filter(lambda x: x.is_standard(), d_hyp))
sage: n_odd + n_hyp == rcc.gamma_std([5])
True

surface_dynamics.interval_exchanges.rauzy_class_cardinality.marking_iterator(profile, left=None, standard=False)

Returns the marked profile associated to a partition

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc
sage: p = Partition([3,2,2])
sage: list(rcc.marking_iterator(p))
[(1, 2, 0),
 (1, 2, 1),
 (1, 3, 0),
 (1, 3, 1),
 (1, 3, 2),
 (2, 2, 2),
 (2, 2, 3),
 (2, 3, 2)]

surface_dynamics.interval_exchanges.rauzy_class_cardinality.number_of_irreducible_permutations(profile=None, marking=None)

Number of permutations for the given profile and marking

INPUT:

• profile - an integer partition such that its sum plus its length is congruent to 0 modulo 2

• markings - None, an element of the profile or a 3-tuple

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc

The connected strata in genus 3:

sage: from surface_dynamics import Stratum

sage: c = Stratum([1,1,1,1], k=1).unique_component()
sage: c.rauzy_diagram()
Rauzy diagram with 1255 permutations
sage: rcc.gamma_irr([2,2,2,2])
1255

sage: c = Stratum([2,1,1], k=1).unique_component()
sage: c.rauzy_diagram()
Rauzy diagram with 2177 permutations
sage: rcc.gamma_irr([3,2,2])
2177

sage: c = Stratum([3,1], k=1).unique_component()
sage: c.rauzy_diagram()
Rauzy diagram with 770 permutations
sage: rcc.gamma_irr([4,2])
770

The non connected strata in genus 3:

sage: c_odd = Stratum([2,2], k=1).odd_component()
sage: c_hyp = Stratum([2,2], k=1).hyperelliptic_component()
sage: c_odd.rauzy_diagram()
Rauzy diagram with 294 permutations
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 63 permutations
sage: rcc.gamma_irr([3,3]) == 294 + 63
True

sage: c_odd = Stratum([4], k=1).odd_component()
sage: c_hyp = Stratum([4], k=1).hyperelliptic_component()
sage: c_odd.rauzy_diagram()
Rauzy diagram with 134 permutations
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 31 permutations
sage: rcc.gamma_irr([5]) == 134 + 31
True

surface_dynamics.interval_exchanges.rauzy_class_cardinality.number_of_standard_permutations(profile, marking=None)

Return the number of standard permutations of given profile

INPUT:

• profile - an integer partition such that the its sum plus its length is congruent to 0 modulo 2

• marking - either None, an element of the profile or a 3-tuple (1, n1, a) or (2, n1, n2)

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc

A ValueError is raised if the partition does not satisfy the requirement:

sage: rcc.gamma_std([5,2])
Traceback (most recent call last):
...
ValueError: the sum of the profile (=[5, 2]) plus its length must be congruent to 0 modulo 2

The Rauzy classes associated to connected strata in genus 3:

sage: from surface_dynamics import Stratum
sage: cc = Stratum([1,1,1,1], k=1).unique_component()
sage: d = cc.rauzy_diagram()
sage: d
Rauzy diagram with 1255 permutations
sage: sum(1 for _ in filter(lambda x: x.is_standard(), d)) == rcc.gamma_std([2,2,2,2])
True

sage: cc = Stratum([2,1,1], k=1).unique_component()
sage: d = cc.rauzy_diagram()
sage: d
Rauzy diagram with 2177 permutations
sage: sum(1 for _ in filter(lambda x: x.is_standard(), d)) == rcc.gamma_std([3,2,2])
True

sage: cc = Stratum([3,1], k=1).unique_component()
sage: d = cc.rauzy_diagram()
sage: d
Rauzy diagram with 770 permutations
sage: sum(1 for _ in filter(lambda x: x.is_standard(), d)) == rcc.gamma_std([4,2])
True

The non connected strata in genus 3:

sage: cc_odd = Stratum([2,2], k=1).odd_component()
sage: cc_hyp = Stratum([2,2], k=1).hyperelliptic_component()
sage: d_odd = cc_odd.rauzy_diagram()
sage: d_hyp = cc_hyp.rauzy_diagram()
sage: d_odd
Rauzy diagram with 294 permutations
sage: d_hyp
Rauzy diagram with 63 permutations
sage: n_odd = sum(1 for _ in filter(lambda x: x.is_standard(), d_odd))
sage: n_hyp = sum(1 for _ in filter(lambda x: x.is_standard(), d_hyp))
sage: n_odd + n_hyp == rcc.gamma_std([3,3])
True

sage: cc_odd = Stratum([4], k=1).odd_component()
sage: cc_hyp = Stratum([4], k=1).hyperelliptic_component()
sage: d_odd = cc_odd.rauzy_diagram()
sage: d_hyp = cc_hyp.rauzy_diagram()
sage: d_odd
Rauzy diagram with 134 permutations
sage: d_hyp
Rauzy diagram with 31 permutations
sage: n_odd = sum(1 for _ in filter(lambda x: x.is_standard(), d_odd))
sage: n_hyp = sum(1 for _ in filter(lambda x: x.is_standard(), d_hyp))
sage: n_odd + n_hyp == rcc.gamma_std([5])
True

surface_dynamics.interval_exchanges.rauzy_class_cardinality.spin_difference_for_irreducible_permutations(profile, marking=None)

Spin difference for the given profile and marking

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc

The non connected strata in genus 3:

sage: from surface_dynamics import Stratum

sage: c_odd = Stratum([2,2], k=1).odd_component()
sage: c_hyp = Stratum([2,2], k=1).hyperelliptic_component()
sage: c_odd.rauzy_diagram()
Rauzy diagram with 294 permutations
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 63 permutations
sage: rcc.delta_irr([3,3]) == 294 - 63
True

sage: c_odd = Stratum([4], k=1).odd_component()
sage: c_hyp = Stratum([4], k=1).hyperelliptic_component()
sage: c_odd.rauzy_diagram()
Rauzy diagram with 134 permutations
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 31 permutations
sage: rcc.delta_irr([5]) == 134 - 31
True

A non connected strata in genus 4:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rdc
sage: a = Stratum([6], k=1)
sage: c_hyp = a.hyperelliptic_component()
sage: c_odd = a.odd_component()
sage: c_even = a.even_component()
sage: c_hyp.rauzy_diagram()
Rauzy diagram with 127 permutations
sage: c_hyp.rauzy_class_cardinality()
127
sage: c_odd.rauzy_diagram()
Rauzy diagram with 5209 permutations
sage: c_odd.rauzy_class_cardinality()
5209
sage: c_even = a.even_component()
sage: c_even.rauzy_diagram()
Rauzy diagram with 2327 permutations
sage: c_even.rauzy_class_cardinality()
2327
sage: 5209 - 2327 - 127
2755
sage: rdc.delta_irr([7])
2755

An example with a very big Rauzy class:

sage: c = Stratum([6,6], k=1).odd_component()
sage: c.rauzy_class_cardinality()
11609364656

surface_dynamics.interval_exchanges.rauzy_class_cardinality.spin_difference_for_standard_permutations(profile, marking=None)

Return the difference odd-even in the given stratum

INPUT:

• p - partition with odd terms

• marking - a 3-tuple (1, n1, a) or (2, n1, n2)

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc

A ValueError is raised if the partition does not fulfill the requirement:

sage: rcc.delta_std([5,2])
Traceback (most recent call last):
...
ValueError: the profile (=[5, 2]) must contain only odd numbers

Non connected strata in genus 3 has two connected components distinguished by their spin parity:

sage: from surface_dynamics import Stratum
sage: cc_odd = Stratum([2,2], k=1).odd_component()
sage: cc_hyp = Stratum([2,2], k=1).hyperelliptic_component()
sage: d_odd = cc_odd.rauzy_diagram()
sage: d_hyp = cc_hyp.rauzy_diagram()
sage: d_odd
Rauzy diagram with 294 permutations
sage: d_hyp
Rauzy diagram with 63 permutations
sage: n_odd = sum(1 for _ in filter(lambda x: x.is_standard(), d_odd))
sage: n_hyp = sum(1 for _ in filter(lambda x: x.is_standard(), d_hyp))
sage: n_odd - n_hyp == rcc.delta_std([3,3])
True

sage: cc_odd = Stratum([4], k=1).odd_component()
sage: cc_hyp = Stratum([4], k=1).hyperelliptic_component()
sage: d_odd = cc_odd.rauzy_diagram()
sage: d_hyp = cc_hyp.rauzy_diagram()
sage: d_odd
Rauzy diagram with 134 permutations
sage: d_hyp
Rauzy diagram with 31 permutations
sage: n_odd = sum(1 for _ in filter(lambda x: x.is_standard(), d_odd))
sage: n_hyp = sum(1 for _ in filter(lambda x: x.is_standard(), d_hyp))
sage: n_odd - n_hyp == rcc.delta_std([5])
True

surface_dynamics.interval_exchanges.rauzy_class_cardinality.split(p, k, i=0)

Splits the i-th term of p into two parts of size k and n-k-1

There is a symmetry split(p, k, i) = split(p, p[i]-k-1, i)

INPUT:

• p - a partition

• k - an integer between 2 and p[i]

• i - integer - the index of the element to split

OUTPUT: a partition

EXAMPLES:

sage: import surface_dynamics.interval_exchanges.rauzy_class_cardinality as rcc
sage: p = Partition([5,1])
sage: rcc.split(p,1,0)
[3, 1, 1]
sage: rcc.split(p,2,0)
[2, 2, 1]
sage: rcc.split(p,3,0)
[3, 1, 1]