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Mp00317: Integer partitions odd partsBinary words
Mp00096: Binary words Foata bijectionBinary words
Mp00136: Binary words rotate back-to-frontBinary words
St000875: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1 => 1 => 1 => 0
[2]
=> 0 => 0 => 0 => 0
[1,1]
=> 11 => 11 => 11 => 0
[3]
=> 1 => 1 => 1 => 0
[2,1]
=> 01 => 01 => 10 => 1
[1,1,1]
=> 111 => 111 => 111 => 0
[4]
=> 0 => 0 => 0 => 0
[3,1]
=> 11 => 11 => 11 => 0
[2,2]
=> 00 => 00 => 00 => 0
[2,1,1]
=> 011 => 011 => 101 => 1
[1,1,1,1]
=> 1111 => 1111 => 1111 => 0
[5]
=> 1 => 1 => 1 => 0
[4,1]
=> 01 => 01 => 10 => 1
[3,2]
=> 10 => 10 => 01 => 0
[3,1,1]
=> 111 => 111 => 111 => 0
[2,2,1]
=> 001 => 001 => 100 => 1
[2,1,1,1]
=> 0111 => 0111 => 1011 => 1
[1,1,1,1,1]
=> 11111 => 11111 => 11111 => 0
[6]
=> 0 => 0 => 0 => 0
[5,1]
=> 11 => 11 => 11 => 0
[4,2]
=> 00 => 00 => 00 => 0
[4,1,1]
=> 011 => 011 => 101 => 1
[3,3]
=> 11 => 11 => 11 => 0
[3,2,1]
=> 101 => 101 => 110 => 1
[3,1,1,1]
=> 1111 => 1111 => 1111 => 0
[2,2,2]
=> 000 => 000 => 000 => 0
[2,2,1,1]
=> 0011 => 0011 => 1001 => 1
[2,1,1,1,1]
=> 01111 => 01111 => 10111 => 1
[1,1,1,1,1,1]
=> 111111 => 111111 => 111111 => 0
[7]
=> 1 => 1 => 1 => 0
[6,1]
=> 01 => 01 => 10 => 1
[5,2]
=> 10 => 10 => 01 => 0
[5,1,1]
=> 111 => 111 => 111 => 0
[4,3]
=> 01 => 01 => 10 => 1
[4,2,1]
=> 001 => 001 => 100 => 1
[4,1,1,1]
=> 0111 => 0111 => 1011 => 1
[3,3,1]
=> 111 => 111 => 111 => 0
[3,2,2]
=> 100 => 010 => 001 => 0
[3,2,1,1]
=> 1011 => 1011 => 1101 => 1
[3,1,1,1,1]
=> 11111 => 11111 => 11111 => 0
[2,2,2,1]
=> 0001 => 0001 => 1000 => 1
[2,2,1,1,1]
=> 00111 => 00111 => 10011 => 1
[2,1,1,1,1,1]
=> 011111 => 011111 => 101111 => 1
[1,1,1,1,1,1,1]
=> 1111111 => 1111111 => 1111111 => 0
[8]
=> 0 => 0 => 0 => 0
[7,1]
=> 11 => 11 => 11 => 0
[6,2]
=> 00 => 00 => 00 => 0
[6,1,1]
=> 011 => 011 => 101 => 1
[5,3]
=> 11 => 11 => 11 => 0
[5,2,1]
=> 101 => 101 => 110 => 1
Description
The semilength of the longest Dyck word in the Catalan factorisation of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the semilength of the longest Dyck word in this factorisation.