searching the database
Your data matches 217 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000175
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2]
=> [1,1]
=> [1]
=> 0
[1,2,3] => [1,1,1]
=> [2,1]
=> [1]
=> 0
[1,3,2] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [2,2]
=> [2]
=> 0
[1,2,4,3] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [3,1]
=> [3,1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [3,1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,3,1,4] => [3,1]
=> [3,1]
=> [1]
=> 0
[2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,4,3,1] => [3,1]
=> [3,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [3,1]
=> [1]
=> 0
[3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,2,1,4] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [3,1]
=> [1]
=> 0
[3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,3,2] => [3,1]
=> [3,1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [3,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [4,1]
=> [1]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [4,1]
=> [1]
=> 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [4,1]
=> [1]
=> 0
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St001549
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 18%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 18%
Values
[2,1] => [2]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,2,3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[1,3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[2,1,3] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[3,2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[1,3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[1,4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[2,3,1,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[2,3,4,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[2,4,1,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[2,4,3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[3,1,2,4] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[3,1,4,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[3,2,4,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[3,4,2,1] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[4,1,2,3] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[4,1,3,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[4,2,1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 0
[4,3,1,2] => [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,3,2,5,4] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,3,4,5,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[1,3,5,2,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,4,2,5,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,4,5,2,3] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0
[1,4,5,3,2] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[1,5,2,3,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 0
[1,5,4,2,3] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => 0
[1,5,4,3,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => ? = 0
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,2,4,5,6,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,2,4,6,3,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,2,5,3,6,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,2,5,6,4,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,2,6,3,4,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,2,6,5,3,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,3,4,2,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,3,4,5,2,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,3,4,5,6,2] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[1,3,4,6,2,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[1,3,4,6,5,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,3,5,2,4,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,3,5,2,6,4] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[1,3,5,4,2,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,3,5,4,6,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,3,5,6,4,2] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[1,3,6,2,4,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[1,3,6,2,5,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,3,6,4,2,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,3,6,4,5,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,3,6,5,2,4] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[1,4,2,3,5,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,4,2,5,3,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,4,2,5,6,3] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[1,4,2,6,3,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[1,4,2,6,5,3] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 2
[1,4,3,5,2,6] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,4,3,5,6,2] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,4,3,6,2,5] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,4,3,6,5,2] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 1
[1,4,5,3,2,6] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 0
[1,4,5,3,6,2] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
[1,4,5,6,2,3] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,1,2,3,4,7,5] => ? = 0
Description
The number of restricted non-inversions between exceedances.
This is for a permutation $\sigma$ of length $n$ given by
$$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$$
Matching statistic: St001001
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[2,1] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[2,4,3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[3,2,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[4,1,3,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,5,4,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
[3,4,1,2,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001371
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001371: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 0
[1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[2,4,3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,2,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[4,1,3,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,5,2,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,4,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,4,1,2,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 2
Description
The length of the longest Yamanouchi prefix of a binary word.
This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Matching statistic: St001730
(load all 59 compositions to match this statistic)
(load all 59 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001730: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 0
[1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 0
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[2,4,3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[3,2,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[4,1,3,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,4,5,2,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,5,4,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
[3,4,1,2,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 2
Description
The number of times the path corresponding to a binary word crosses the base line.
Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001803
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 0
[1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 0
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[2,4,3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[3,2,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[4,1,3,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
[4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,5,4,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
[3,4,1,2,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 2
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001195
(load all 67 compositions to match this statistic)
(load all 67 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[2,1] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 0 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[2,4,3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[3,2,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[4,1,3,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,4,1,2,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 0 + 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 2 + 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001208
(load all 76 compositions to match this statistic)
(load all 76 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 1 = 0 + 1
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 0 + 1
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[2,4,3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[3,2,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[4,1,3,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
[4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => 1 = 0 + 1
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,1,4,5,6,7,2] => ? = 1 + 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,4,5,2,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,5,4,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
[3,4,1,2,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 0 + 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0 + 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 2 + 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St001804
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 9%
Values
[2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 2 = 0 + 2
[1,2,3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 2 = 0 + 2
[1,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 0 + 2
[2,1,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 0 + 2
[3,2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 2 = 0 + 2
[1,2,3,4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 0 + 2
[1,2,4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[1,3,2,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[1,3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
[1,4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
[1,4,3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[2,1,3,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[2,3,1,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
[2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[2,4,3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
[3,1,2,4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
[3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[3,2,1,4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[3,2,4,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
[3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[4,1,3,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
[4,2,1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 2 = 0 + 2
[4,2,3,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 2 = 0 + 2
[4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0 + 2
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,3,4,5,6,7],[2,8,9,10,11,12]]
=> ? = 1 + 2
[1,2,3,5,4] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,2,4,3,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,2,4,5,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,2,5,3,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,2,5,4,3] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,3,2,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,3,2,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 0 + 2
[1,3,4,2,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,3,4,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,3,5,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,3,5,4,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,4,2,3,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,4,2,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,4,3,2,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,4,3,5,2] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,4,5,2,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 0 + 2
[1,4,5,3,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,5,2,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,5,2,4,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,5,3,2,4] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[1,5,3,4,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[1,5,4,2,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,5,4,3,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 0 + 2
[2,1,3,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[2,1,3,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 0 + 2
[2,1,4,3,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 0 + 2
[2,1,4,5,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[2,1,5,3,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[2,1,5,4,3] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 0 + 2
[2,3,1,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[2,3,1,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[2,3,4,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,3,5,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,4,1,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,4,3,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[2,4,3,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,4,5,1,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[2,5,1,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,5,3,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[2,5,3,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[2,5,4,3,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,1,2,4,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[3,1,2,5,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,1,4,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,1,5,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,2,1,4,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[3,2,1,5,4] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 0 + 2
[3,2,4,1,5] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[3,2,4,5,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,2,5,1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,2,5,4,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 2 = 0 + 2
[3,4,1,2,5] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 2 = 0 + 2
[3,4,1,5,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,4,2,1,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[3,4,5,2,1] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,5,1,2,4] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 2 = 0 + 2
[3,5,2,4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,1,2,3,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,1,3,5,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,2,1,5,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,2,3,1,5] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[4,2,5,3,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,3,1,2,5] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[4,5,3,2,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,1,2,4,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,1,3,2,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,2,1,3,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,2,3,4,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0 + 2
[5,2,4,1,3] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,3,1,4,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[5,4,3,1,2] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 0 + 2
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0 + 2
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 2 + 2
Description
The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
This statistic equals $\max_C\big(\ell(C) - \ell(T)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
Matching statistic: St001301
(load all 87 compositions to match this statistic)
(load all 87 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 9%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 0% ●values known / values provided: 0%●distinct values known / distinct values provided: 9%
Values
[2,1] => [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 0
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 0
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ? = 0
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
=> ? = 0
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 0
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
Description
The first Betti number of the order complex associated with the poset.
The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.
The following 207 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001256Number of simple reflexive modules that are 2-stable reflexive. St001890The maximum magnitude of the Möbius function of a poset. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000879The number of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St000667The greatest common divisor of the parts of the partition. St000993The multiplicity of the largest part of an integer partition. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000781The number of proper colouring schemes of a Ferrers diagram. St001128The exponens consonantiae of a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001895The oddness of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St001889The size of the connectivity set of a signed permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001625The Möbius invariant of a lattice. St000068The number of minimal elements in a poset. St001429The number of negative entries in a signed permutation. St001621The number of atoms of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001964The interval resolution global dimension of a poset. St000627The exponent of a binary word. St000878The number of ones minus the number of zeros of a binary word. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St000699The toughness times the least common multiple of 1,. St000948The chromatic discriminant of a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001281The normalized isoperimetric number of a graph. St001305The number of induced cycles on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000636The hull number of a graph. St001029The size of the core of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001654The monophonic hull number of a graph. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001768The number of reduced words of a signed permutation. St000074The number of special entries. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000366The number of double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000629The defect of a binary word. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000787The number of flips required to make a perfect matching noncrossing. St000877The depth of the binary word interpreted as a path. St000894The trace of an alternating sign matrix. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001381The fertility of a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000056The decomposition (or block) number of a permutation. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000942The number of critical left to right maxima of the parking functions. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001410The minimal entry of a semistandard tableau. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001884The number of borders of a binary word. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000084The number of subtrees. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000328The maximum number of child nodes in a tree. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000905The number of different multiplicities of parts of an integer composition. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001330The hat guessing number of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000907The number of maximal antichains of minimal length in a poset. St000264The girth of a graph, which is not a tree. St000782The indicator function of whether a given perfect matching is an L & P matching.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!