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Matching statistic: St000876
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Mapping
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 2
01 => 2
10 => 1
11 => 2
000 => 3
001 => 3
010 => 2
011 => 3
100 => 1
101 => 2
110 => 1
111 => 3
0000 => 4
0001 => 4
0010 => 3
0011 => 4
0100 => 2
0101 => 3
0110 => 2
0111 => 4
1000 => 2
1001 => 2
1010 => 1
1011 => 3
1100 => 1
1101 => 2
1110 => 2
1111 => 4
00000 => 5
00001 => 5
00010 => 4
00011 => 5
00100 => 3
00101 => 4
00110 => 3
00111 => 5
01000 => 3
01001 => 3
01010 => 2
01011 => 4
01100 => 2
01101 => 3
01110 => 3
01111 => 5
10000 => 3
10001 => 3
10010 => 2
10011 => 3
Description
The number of factors in the Catalan decomposition 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 number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.