Shuffle product of iterables

The shuffle product of two sequences of lengths \(m\) and \(n\) is a sum over the \(\binom{m+n}{n}\) ways of interleaving the two sequences.

That could be defined inductively by:

\[(a_n)_{n \geqslant 0} \Cup (b_m)_{m \geqslant 0} = a_0 \cdot \left((a_n)_{n \geqslant 1} \Cup (b_m)_{m \geqslant 0}\right) + b_0 \cdot \left((a_n)_{n \geqslant 0} \Cup (b_m)_{m \geqslant 1}\right)\]

with \((a_n)\) and \((b_m)\) two non-empty sequences and if one of them is empty then the product is equals to the other.

The shuffle product has been introduced by S. Eilenberg and S. Mac Lane in 1953 [EilLan53].

EXAMPLE:

sage: from sage.combinat.shuffle import ShuffleProduct
sage: list(ShuffleProduct([1,2], ["a", "b", "c"]))
[[1, 2, 'a', 'b', 'c'],
 ['a', 1, 2, 'b', 'c'],
 [1, 'a', 2, 'b', 'c'],
 ['a', 'b', 1, 2, 'c'],
 ['a', 1, 'b', 2, 'c'],
 [1, 'a', 'b', 2, 'c'],
 ['a', 'b', 'c', 1, 2],
 ['a', 'b', 1, 'c', 2],
 ['a', 1, 'b', 'c', 2],
 [1, 'a', 'b', 'c', 2]]

References:

[EilLan53]On the groups \(H(\pi, n)\), I, Samuel Eilenberg and Saunders Mac Lane, 1953.

Author:

  • Jean-Baptiste Priez
class sage.combinat.shuffle.SetShuffleProduct(l1, l2, element_constructor=None)

Bases: sage.structure.sage_object.SageObject

The union of all possible shuffle products of two sets of iterables.

TESTS:

sage: from sage.combinat.shuffle import SetShuffleProduct
sage: TestSuite(SetShuffleProduct).run()

EXAMPLE:

sage: from sage.combinat.shuffle import SetShuffleProduct
sage: sorted(SetShuffleProduct({(1,), (2,3)}, {(4,5), (6,)}))
[[1, 4, 5],
 [1, 6],
 [2, 3, 4, 5],
 [2, 3, 6],
 [2, 4, 3, 5],
 [2, 4, 5, 3],
 [2, 6, 3],
 [4, 1, 5],
 [4, 2, 3, 5],
 [4, 2, 5, 3],
 [4, 5, 1],
 [4, 5, 2, 3],
 [6, 1],
 [6, 2, 3]]
cardinality()

The cardinality is defined by the sum of the cardinality of all shuffles. That means by a sum of binomials.

TESTS:

sage: from sage.combinat.shuffle import SetShuffleProduct
sage: SetShuffleProduct([[1,2],[3,4]], [[1,4]], element_constructor=set).cardinality()
12
class sage.combinat.shuffle.ShuffleProduct(l1, l2, element_constructor=None)

Bases: sage.structure.sage_object.SageObject

Shuffle product of two iterable.

EXAMPLES:

sage: from sage.combinat.shuffle import ShuffleProduct
sage: list(ShuffleProduct("abc", "de", element_constructor="".join))
['abcde',
 'adbce',
 'dabce',
 'abdce',
 'adebc',
 'daebc',
 'deabc',
 'adbec',
 'dabec',
 'abdec']
sage: list(ShuffleProduct("", "de", element_constructor="".join))
['de']
cardinality()

Return the number of shuffles of \(l_1\) and \(l_2\), respectively of lengths \(m\) and \(n\), which is \(\binom{m+n}{n}\).

TESTS:

sage: from sage.combinat.shuffle import ShuffleProduct
sage: ShuffleProduct([3,1,2], [4,2,1,3]).cardinality()
35
sage: ShuffleProduct([3,1,2,5,6,4], [4,2,1,3]).cardinality() == binomial(10,4)
True

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