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Find the rank of a totally balanced binary sequence

Contributed by: Ed Pegg Jr


returns the position of the given totally balanced binary sequence bin with n ones, in a particular enumeration of all such balanced sequences.

Details and Options

A binary sequence is considered totally balanced if the number of zeros is at least as large as the number of ones as one progresses from left to right in a list of zeros and ones, and the total counts are equal (implying the first element must be zero and the last element one).
For n ones there are Cn totally balanced binary sequences, where Cn is the Catalan number.
The value returned is an index into a certain ordering of the set of all possible balanced sequences with n zeros and n ones, with indexing starting at 0.
The indexing scheme can be inverted using the resource function CatalanUnrank.
Brackets in a computer program must be balanced. One can think of a proper bracketing as having left brackets corresponding to zeros and right brackets to ones in a balanced binary sequence.


Basic Examples (2) 

Find the rank of a given balanced binary sequence:

ResourceFunction["CatalanRank"][5, {0, 0, 1, 0, 1, 0, 1, 1, 0, 1}]

Find all totally balanced binary sequences of length 10:

balanced = Select[Permutations[Join[Table[0, 5], Table[1, 5]]], Min[Table[
      Count[Take[#, n], 0] - Count[Take[#, n], 1], {n, 1, 10}]] == 0 &]

Find the Catalan rank of each balanced sequence:

ResourceFunction["CatalanRank"][5, #] & /@ balanced

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