Basic Examples (1) 

Stuffle product of two words with an inert head:

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Scope (5) 

Expand the 3-way stuffle of words:

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Expand a 2 × 3 stuffle:

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Treat the empty word as the identity:

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StuffleProductExpand distributes an expression linearly before expanding:

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Stuffle independently by head and multiply the results:

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Properties and Relations (6) 

Stuffle product expansion is commutative:

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Stuffle product expansion is associative:

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If the inputs have lengths m and n, every output term has length m+n-k where 0≤k≤Min[m,n]:

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Relate term count to Delannoy number and Multinomial:

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The number of terms with exactly k merges (output length m+n-k), for k=0, 1, …, Min[m,n] is Multinomial[Max[m,n]-k, Min[m,n]-k, k]:

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Expanding an already stuffled sum changes nothing (indempotence):

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Reverse both inputs to reverse every shuffled word (reversal symmetry):

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Possible Issues (2) 

Bare lists multiply element-wise:

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Wrap as h[{…}] or use Inactive[List] to treat them as words before expanding:

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