Shuffle Product

It interleaves two sequences without changing either sequence’s internal order.

For ab and xy, possible results include abxy, axby, and xyab; every result keeps a before b and x before y. The operation adds all valid interleavings as terms. If repeated letters make different choices produce the same word, that word gets a coefficient.

A documented line of mathematical work includes shuffle maps studied by mathematicians Eilenberg and Mac Lane in 1953. Around 1954, Chen applied the structure to iterated path integrals—repeated integrals built along a path—and Ree later developed its algebraic role.

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