uconf.models.barratt_eccles

Barratt–Eccles operad model on sequences of permutations.

Classes

BarrattEccles(n, base_ring)

Barratt--Eccles operad component in fixed arity.

class uconf.models.barratt_eccles.BarrattEccles(n, base_ring)

Bases: CombinatorialFreeModule

Barratt–Eccles operad component in fixed arity.

Basis elements are tuples of permutations in S_n with no consecutive duplicates. Degenerate inputs map to zero, while malformed inputs raise.

EXAMPLES:

sage: from sage.all import QQ
sage: from uconf import BarrattEccles
sage: E2 = BarrattEccles(2, QQ)
sage: x = E2(((1, 2), (2, 1)))
sage: x.arity()
2
sage: E2.degree_on_basis(next(iter(x.support())))
1

connectivity: ClassVar[int] = 0

All components live in non-negative degrees.

__init__(n, base_ring)

Initialize E_n over base_ring.

rho(data)

Build the normalized element (id, sigma_1, sigma_1 sigma_2, ...).

Parameters:

data (tuple | list)

Return type:

Element

arity()

Return the fixed arity of this operad component.

Return type:

int

static unit(base_ring)

Return the operadic unit in arity 1.

Return type:

Element

static unit_key()

Return the basis key of the unit element in arity 1.

Return type:

tuple

planar_basis_iter(d)

Iterate over planar basis elements in degree d.

Parameters:

d (int)

Return type:

Iterator[Element]

basis_iter(d)

Iterate over all basis elements in degree d.

Parameters:

d (int)

Return type:

Iterator[Element]

graded_basis

/home/runner/work/uconf/uconf/src/uconf/models/barratt_eccles.py (starting at line 191) Return the Family of all basis elements in degree d.

Type:

File

graded_planar_basis

/home/runner/work/uconf/uconf/src/uconf/models/barratt_eccles.py (starting at line 196) Return the Family of planar basis elements in degree d.

Type:

File

degree_on_basis(element)

Return homological degree of a basis element.

Parameters:

element (tuple)

Return type:

int

static compose(x, i, y)

Operadic composition x \circ_i y using shuffle/EZ lifting.

Parameters

x, y:

Barratt–Eccles elements over the same base ring.

i:

Input slot of x into which y is inserted.

Returns

BarrattEccles.Element

The composite in arity x.arity() + y.arity() - 1.

EXAMPLES:

sage: from sage.all import QQ
sage: from uconf import BarrattEccles
sage: x = BarrattEccles(2, QQ)(((1, 2),))
sage: y = BarrattEccles(2, QQ)(((1, 2), (2, 1)))
sage: BarrattEccles.compose(x, 1, y).arity()
3

Parameters:

• x (BarrattEccles)

• i (int)

• y (BarrattEccles)

Return type:

BarrattEccles

BarrattEccles.Element

alias of Element