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Matching statistic: St000885
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St000885: 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 => 0
11 => 2
000 => 3
001 => 3
010 => 1
011 => 3
100 => 1
101 => 1
110 => 1
111 => 3
0000 => 4
0001 => 4
0010 => 2
0011 => 4
0100 => 2
0101 => 2
0110 => 2
0111 => 4
1000 => 2
1001 => 2
1010 => 0
1011 => 2
1100 => 0
1101 => 2
1110 => 2
1111 => 4
00000 => 5
00001 => 5
00010 => 3
00011 => 5
00100 => 3
00101 => 3
00110 => 3
00111 => 5
01000 => 3
01001 => 3
01010 => 1
01011 => 3
01100 => 1
01101 => 3
01110 => 3
01111 => 5
10000 => 3
10001 => 3
10010 => 1
10011 => 3
Description
The number of critical steps 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 critical steps $\ell + m$ in the Catalan factorisation.
The distribution of this statistic on words of length $n$ is
$$
(n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}.
$$
Matching statistic: St001880
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
1 => [1] => [[1],[]]
=> ([],1)
=> ? ∊ {1,1}
00 => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {0,2,2,2}
01 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,2,2,2}
10 => [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? ∊ {0,2,2,2}
11 => [2] => [[2],[]]
=> ([(0,1)],2)
=> ? ∊ {0,2,2,2}
000 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
001 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1}
010 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
011 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {1,1,1,1}
100 => [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? ∊ {1,1,1,1}
101 => [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
110 => [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,1}
111 => [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
0000 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
0001 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
0011 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
0100 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
0110 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
0111 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
1000 => [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
1001 => [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
1011 => [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
1100 => [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
1110 => [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {0,0,2,2,2,2,2,2,2,2,2,4}
1111 => [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
00000 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
00001 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
00011 => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
00111 => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
01000 => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
01100 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
01110 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
01111 => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
10000 => [1,4] => [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
10001 => [1,3,1] => [[3,3,1],[2]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
10011 => [1,2,2] => [[3,2,1],[1]]
=> ([(0,3),(0,4),(1,2),(1,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
10111 => [1,1,3] => [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
11000 => [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
11011 => [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
11100 => [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
11101 => [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
11110 => [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5}
11111 => [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
000000 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
010101 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
101010 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
111111 => [6] => [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
0000000 => [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
0101010 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
1010101 => [1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
1111111 => [7] => [[7],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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