- FindStat

Your data matches 72 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000175

Mapping

Mp00166: Signed permutations —even cycle type⟶ 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

[1,2] => [1,1]
 => [1]
 => 0
[1,2,3] => [1,1,1]
 => [1,1]
 => 0
[1,2,-3] => [1,1]
 => [1]
 => 0
[1,-2,3] => [1,1]
 => [1]
 => 0
[-1,2,3] => [1,1]
 => [1]
 => 0
[1,3,2] => [2,1]
 => [1]
 => 0
[1,-3,-2] => [2,1]
 => [1]
 => 0
[2,1,3] => [2,1]
 => [1]
 => 0
[-2,-1,3] => [2,1]
 => [1]
 => 0
[3,2,1] => [2,1]
 => [1]
 => 0
[-3,2,-1] => [2,1]
 => [1]
 => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 0
[1,2,3,-4] => [1,1,1]
 => [1,1]
 => 0
[1,2,-3,4] => [1,1,1]
 => [1,1]
 => 0
[1,2,-3,-4] => [1,1]
 => [1]
 => 0
[1,-2,3,4] => [1,1,1]
 => [1,1]
 => 0
[1,-2,3,-4] => [1,1]
 => [1]
 => 0
[1,-2,-3,4] => [1,1]
 => [1]
 => 0
[-1,2,3,4] => [1,1,1]
 => [1,1]
 => 0
[-1,2,3,-4] => [1,1]
 => [1]
 => 0
[-1,2,-3,4] => [1,1]
 => [1]
 => 0
[-1,-2,3,4] => [1,1]
 => [1]
 => 0
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 0
[1,2,4,-3] => [1,1]
 => [1]
 => 0
[1,2,-4,3] => [1,1]
 => [1]
 => 0
[1,2,-4,-3] => [2,1,1]
 => [1,1]
 => 0
[1,-2,4,3] => [2,1]
 => [1]
 => 0
[1,-2,-4,-3] => [2,1]
 => [1]
 => 0
[-1,2,4,3] => [2,1]
 => [1]
 => 0
[-1,2,-4,-3] => [2,1]
 => [1]
 => 0
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 0
[1,3,2,-4] => [2,1]
 => [1]
 => 0
[1,3,-2,4] => [1,1]
 => [1]
 => 0
[1,-3,2,4] => [1,1]
 => [1]
 => 0
[1,-3,-2,4] => [2,1,1]
 => [1,1]
 => 0
[1,-3,-2,-4] => [2,1]
 => [1]
 => 0
[-1,3,2,4] => [2,1]
 => [1]
 => 0
[-1,-3,-2,4] => [2,1]
 => [1]
 => 0
[1,3,4,2] => [3,1]
 => [1]
 => 0
[1,3,-4,-2] => [3,1]
 => [1]
 => 0
[1,-3,4,-2] => [3,1]
 => [1]
 => 0
[1,-3,-4,2] => [3,1]
 => [1]
 => 0
[1,4,2,3] => [3,1]
 => [1]
 => 0
[1,4,-2,-3] => [3,1]
 => [1]
 => 0
[1,-4,2,-3] => [3,1]
 => [1]
 => 0
[1,-4,-2,3] => [3,1]
 => [1]
 => 0
[1,4,3,2] => [2,1,1]
 => [1,1]
 => 0
[1,4,3,-2] => [1,1]
 => [1]
 => 0
[1,4,-3,2] => [2,1]
 => [1]
 => 0
[1,-4,3,2] => [1,1]
 => [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: St000278

Mapping

Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000278: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,2,3] => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,2,-3] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,-2,3] => [1,1]
 => [1]
 => 1 = 0 + 1
[-1,2,3] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,3,2] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,-3,-2] => [2,1]
 => [1]
 => 1 = 0 + 1
[2,1,3] => [2,1]
 => [1]
 => 1 = 0 + 1
[-2,-1,3] => [2,1]
 => [1]
 => 1 = 0 + 1
[3,2,1] => [2,1]
 => [1]
 => 1 = 0 + 1
[-3,2,-1] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 1 = 0 + 1
[1,2,3,-4] => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,2,-3,4] => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,2,-3,-4] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,-2,3,4] => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,-2,3,-4] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,-2,-3,4] => [1,1]
 => [1]
 => 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
 => [1,1]
 => 1 = 0 + 1
[-1,2,3,-4] => [1,1]
 => [1]
 => 1 = 0 + 1
[-1,2,-3,4] => [1,1]
 => [1]
 => 1 = 0 + 1
[-1,-2,3,4] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,2,4,3] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,2,4,-3] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,2,-4,3] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,2,-4,-3] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,-2,4,3] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,-2,-4,-3] => [2,1]
 => [1]
 => 1 = 0 + 1
[-1,2,4,3] => [2,1]
 => [1]
 => 1 = 0 + 1
[-1,2,-4,-3] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,3,2,4] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,3,2,-4] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,3,-2,4] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,-3,2,4] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,-3,-2,4] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,-3,-2,-4] => [2,1]
 => [1]
 => 1 = 0 + 1
[-1,3,2,4] => [2,1]
 => [1]
 => 1 = 0 + 1
[-1,-3,-2,4] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,3,4,2] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,3,-4,-2] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,-3,4,-2] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,-3,-4,2] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,2,3] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,-2,-3] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,-4,2,-3] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,-4,-2,3] => [3,1]
 => [1]
 => 1 = 0 + 1
[1,4,3,2] => [2,1,1]
 => [1,1]
 => 1 = 0 + 1
[1,4,3,-2] => [1,1]
 => [1]
 => 1 = 0 + 1
[1,4,-3,2] => [2,1]
 => [1]
 => 1 = 0 + 1
[1,-4,3,2] => [1,1]
 => [1]
 => 1 = 0 + 1

Description

The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. This is the multinomial of the multiplicities of the parts, see [1]. This is the same as

$m_\lambda(x_1,\dotsc,x_k)$ evaluated at

$x_1=\dotsb=x_k=1$ , where

$k$ is the number of parts of

$\lambda$ . An explicit formula is

$\frac{k!}{m_1(\lambda)! m_2(\lambda)! \dotsb m_k(\lambda) !}$ where

$m_i(\lambda)$ is the number of parts of

$\lambda$ equal to

$i$ .

Matching statistic: St001083

Mapping

Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-3,-2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-2,-1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[3,2,1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-3,2,-1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,2,3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,-2,3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,2,4,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,-4,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,-4,-3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,-2,4,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-2,-4,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,2,4,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,2,-4,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,3,2,-4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,-2,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-3,2,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-3,-2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,-3,-2,-4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,3,2,4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,-3,-2,4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,4,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,3,-4,-2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-3,4,-2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-3,-4,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,-2,-3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-4,2,-3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-4,-2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,4,3,-2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,4,-3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-4,3,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0

Description

The number of boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern

$132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.

Matching statistic: St001535

Mapping

Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St001535: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,2,3] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[1,2,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,-2,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,2,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,-3,-2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[2,1,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[-2,-1,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[3,2,1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[-3,2,-1] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,2,3,4] => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 0
[1,2,3,-4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[1,2,-3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[1,2,-3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,-2,3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[1,-2,3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,-2,-3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,2,3,4] => [1,1,1]
 => [1,1,0,1,0,0]
 => [2,3,1] => 0
[-1,2,3,-4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,2,-3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-1,-2,3,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,2,4,-3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,2,-4,3] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,2,-4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,-2,4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,-2,-4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[-1,2,4,3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[-1,2,-4,-3] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,3,2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,3,-2,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,-3,2,4] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,-3,-2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,-3,-2,-4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[-1,3,2,4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[-1,-3,-2,4] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,3,4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,3,-4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,-3,4,-2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,-3,-4,2] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,4,2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,4,-2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,-4,2,-3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,-4,-2,3] => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 0
[1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 0
[1,4,3,-2] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[1,4,-3,2] => [2,1]
 => [1,0,1,1,0,0]
 => [1,3,2] => 0
[1,-4,3,2] => [1,1]
 => [1,1,0,0]
 => [2,1] => 0
[-3,2,-8,-6,-4,1,5,7] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[5,7,-4,2,3,6,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-4,3,-7,-5,2,6,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-5,2,4,8,-7,1,3,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-6,-4,3,5,-7,2,-8,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[8,-6,2,4,-7,1,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-7,-5,3,8,1,2,4,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-3,2,7,8,-4,1,5,6] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-8,-6,-5,4,2,-7,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-8,6,-7,3,5,1,2,4] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-6,5,-7,4,-8,-2,1,3] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[5,-6,-7,4,-8,-3,1,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[7,8,1,4,-6,-3,2,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-6,2,-5,7,1,8,3,4] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-6,2,-4,1,7,8,3,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[5,2,-8,3,-7,-4,-6,1] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[7,8,1,-3,5,2,-6,4] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-3,-4,-8,7,5,1,-6,2] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[6,1,8,4,-7,-3,2,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-8,-6,3,1,-7,-4,2,5] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0
[-2,-7,1,-6,5,3,-8,4] => [7,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,5,6,8,7] => ? = 0

Description

The number of cyclic alignments of a permutation. The pair

$(i,j)$ is a cyclic alignment of a permutation

$\pi$ if

$i, j, \pi(j), \pi(i)$ are cyclically ordered and all distinct, see Section 5 of [1]

Matching statistic: St001685

Mapping

Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-3,-2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-2,-1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[3,2,1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-3,2,-1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,2,3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,-2,3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,2,4,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,-4,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,-4,-3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,-2,4,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-2,-4,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,2,4,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,2,-4,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,3,2,-4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,-2,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-3,2,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-3,-2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,-3,-2,-4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,3,2,4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,-3,-2,4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,4,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,3,-4,-2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-3,4,-2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-3,-4,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,-2,-3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-4,2,-3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-4,-2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,4,3,-2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,4,-3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-4,3,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,3,4,7,5,6] => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => ? = 0
[1,2,5,3,7,4,6] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,2,7,3,4,5,6] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,2,7,3,6,4,5] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,4,7,2,3,5,6] => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? = 1
[1,5,7,2,3,4,6] => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? = 0
[1,6,7,2,3,4,5] => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => ? = 1
[1,6,7,2,5,3,4] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,6,7,4,2,3,5] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[3,5,7,1,2,4,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? = 0
[3,6,1,2,7,4,5] => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => ? = 0
[3,7,4,1,2,5,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[3,7,4,5,1,2,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[3,7,4,5,6,1,2] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? = 0
[3,7,5,6,1,2,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[3,7,6,1,2,4,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[5,3,4,7,1,2,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? = 0
[5,4,7,1,2,3,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[7,1,3,2,4,5,6] => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? = 0
[7,1,4,6,2,3,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[7,1,5,2,3,4,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? = 0
[7,1,6,4,5,2,3] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[7,2,4,1,3,5,6] => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? = 0
[7,2,5,1,3,4,6] => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? = 1
[7,2,6,1,3,4,5] => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? = 0
[7,3,5,1,2,4,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[7,5,1,4,2,3,6] => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? = 1
[5,6,7,8,3,4,1,2] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 0
[5,6,7,8,3,1,2,4] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => ? = 0
[5,6,7,8,1,4,2,3] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => ? = 0
[7,8,5,6,3,4,1,2] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ? = 0
[6,8,-4,-3,1,2,5,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => ? = 0
[-3,2,-8,-6,-4,1,5,7] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[5,7,-4,2,3,6,-8,1] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[-3,6,8,-5,1,2,4,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => ? = 0
[-4,3,-7,-5,2,6,-8,1] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[-5,2,4,8,-7,1,3,6] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[-4,3,6,-7,-8,-5,1,2] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => ? = 0
[-6,-4,3,5,-7,2,-8,1] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[5,6,3,4,-7,2,-8,1] => [4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8] => ? = 2
[8,-6,2,4,-7,1,3,5] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[4,-8,-6,-3,-7,1,2,5] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 0
[-4,3,-8,-6,-7,1,2,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => ? = 0
[-7,-6,5,8,3,1,2,4] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => ? = 0
[5,-6,3,8,2,-7,1,4] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [4,2,5,1,3,6,7,8] => ? = 1
[-7,-6,2,-5,3,4,-8,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => ? = 0
[-5,2,6,8,3,-7,1,4] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [4,2,5,1,3,6,7,8] => ? = 1
[5,6,7,1,8,2,3,4] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => ? = 0
[8,-7,-6,3,1,2,4,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => ? = 0
[-7,-5,3,8,1,2,4,6] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0

Description

The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.

Matching statistic: St001705

Mapping

Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001705: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-3,-2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-2,-1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[3,2,1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-3,2,-1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,2,3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,-2,3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,2,4,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,-4,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,-4,-3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,-2,4,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-2,-4,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,2,4,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,2,-4,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,3,2,-4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,-2,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-3,2,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-3,-2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,-3,-2,-4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,3,2,4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,-3,-2,4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,4,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,3,-4,-2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-3,4,-2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-3,-4,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,-2,-3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-4,2,-3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-4,-2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,4,3,-2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,4,-3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-4,3,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,3,4,7,5,6] => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => ? = 0
[1,2,5,3,7,4,6] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,2,7,3,4,5,6] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,2,7,3,6,4,5] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,4,7,2,3,5,6] => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? = 1
[1,5,7,2,3,4,6] => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? = 0
[1,6,7,2,3,4,5] => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [5,2,6,7,1,3,4] => ? = 1
[1,6,7,2,5,3,4] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,6,7,4,2,3,5] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[3,5,7,1,2,4,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? = 0
[3,6,1,2,7,4,5] => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [5,6,3,4,1,2,7] => ? = 0
[3,7,4,1,2,5,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[3,7,4,5,1,2,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[3,7,4,5,6,1,2] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? = 0
[3,7,5,6,1,2,4] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[3,7,6,1,2,4,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[5,3,4,7,1,2,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? = 0
[5,4,7,1,2,3,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[7,1,3,2,4,5,6] => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? = 0
[7,1,4,6,2,3,5] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[7,1,5,2,3,4,6] => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [3,4,1,2,5,6,7] => ? = 0
[7,1,6,4,5,2,3] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[7,2,4,1,3,5,6] => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? = 0
[7,2,5,1,3,4,6] => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? = 1
[7,2,6,1,3,4,5] => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [2,1,3,4,5,6,7] => ? = 0
[7,3,5,1,2,4,6] => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [4,5,6,1,2,3,7] => ? = 0
[7,5,1,4,2,3,6] => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [4,2,5,1,3,6,7] => ? = 1
[5,6,7,8,3,4,1,2] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 0
[5,6,7,8,3,1,2,4] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => ? = 0
[5,6,7,8,1,4,2,3] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => ? = 0
[7,8,5,6,3,4,1,2] => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [7,8,5,6,3,4,1,2] => ? = 0
[6,8,-4,-3,1,2,5,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => ? = 0
[-3,2,-8,-6,-4,1,5,7] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[5,7,-4,2,3,6,-8,1] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[-3,6,8,-5,1,2,4,7] => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => [3,4,1,2,5,6,7,8] => ? = 0
[-4,3,-7,-5,2,6,-8,1] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[-5,2,4,8,-7,1,3,6] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[-4,3,6,-7,-8,-5,1,2] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => ? = 0
[-6,-4,3,5,-7,2,-8,1] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[5,6,3,4,-7,2,-8,1] => [4,2,1,1]
 => [[1,4,7,8],[2,6],[3],[5]]
 => [5,3,2,6,1,4,7,8] => ? = 2
[8,-6,2,4,-7,1,3,5] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0
[4,-8,-6,-3,-7,1,2,5] => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => ? = 0
[-4,3,-8,-6,-7,1,2,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => ? = 0
[-7,-6,5,8,3,1,2,4] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => ? = 0
[5,-6,3,8,2,-7,1,4] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [4,2,5,1,3,6,7,8] => ? = 1
[-7,-6,2,-5,3,4,-8,1] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => ? = 0
[-5,2,6,8,3,-7,1,4] => [5,2,1]
 => [[1,3,6,7,8],[2,5],[4]]
 => [4,2,5,1,3,6,7,8] => ? = 1
[5,6,7,1,8,2,3,4] => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [5,6,3,4,1,2,7,8] => ? = 0
[8,-7,-6,3,1,2,4,5] => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => [4,5,6,1,2,3,7,8] => ? = 0
[-7,-5,3,8,1,2,4,6] => [7,1]
 => [[1,3,4,5,6,7,8],[2]]
 => [2,1,3,4,5,6,7,8] => ? = 0

Description

The number of occurrences of the pattern 2413 in a permutation.

Matching statistic: St001811

Mapping

Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 86%●distinct values known / distinct values provided: 67%

Values

[1,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,3] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-3,-2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[2,1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-2,-1,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[3,2,1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-3,2,-1] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,2,3,4] => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,2,3,-4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,2,-3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[1,-2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,3,4] => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 0
[-1,2,3,-4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,2,-3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[-1,-2,3,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,4,3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,2,4,-3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,-4,3] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,-4,-3] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,-2,4,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-2,-4,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,2,4,3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,2,-4,-3] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,3,2,-4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,-2,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-3,2,4] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,-3,-2,4] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,-3,-2,-4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,3,2,4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[-1,-3,-2,4] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,3,4,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,3,-4,-2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-3,4,-2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-3,-4,2] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,-2,-3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-4,2,-3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,-4,-2,3] => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 0
[1,4,3,2] => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 0
[1,4,3,-2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,4,-3,2] => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 0
[1,-4,3,2] => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,2,3,4,5,6] => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => ? = 0
[1,2,3,6,4,5] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => ? = 0
[1,2,5,3,4,6] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => ? = 0
[1,2,5,6,3,4] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => ? = 2
[1,4,2,3,5,6] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => ? = 0
[1,4,2,6,3,5] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[1,4,5,2,3,6] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => ? = 2
[1,4,5,6,2,3] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[1,5,2,3,4,6] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => ? = 0
[1,5,6,2,3,4] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[1,6,2,3,4,5] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[1,6,2,5,3,4] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[1,6,4,2,3,5] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[1,6,4,5,2,3] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[3,1,2,4,5,6] => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => ? = 0
[3,1,2,6,4,5] => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ? = 0
[3,1,5,2,4,6] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[3,1,5,6,2,4] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[3,4,1,2,5,6] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => ? = 2
[3,4,1,6,2,5] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[3,4,5,1,2,6] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[3,4,5,6,1,2] => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ? = 0
[3,5,1,2,4,6] => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => ? = 1
[3,5,6,1,2,4] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[3,6,1,2,4,5] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[3,6,1,5,2,4] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[3,6,4,1,2,5] => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ? = 0
[3,6,4,5,1,2] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[4,1,2,3,5,6] => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => ? = 0
[4,5,1,2,3,6] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[4,5,6,1,2,3] => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [5,6,3,4,1,2] => ? = 0
[4,6,5,1,2,3] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[5,1,2,3,4,6] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[5,1,2,6,3,4] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[5,1,4,2,3,6] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[5,1,4,6,2,3] => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ? = 0
[5,3,1,2,4,6] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[5,3,1,6,2,4] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[5,3,4,1,2,6] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[5,3,4,6,1,2] => [4,2]
 => [[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ? = 0
[5,6,1,2,3,4] => [3,3]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ? = 0
[5,6,1,4,2,3] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[5,6,3,1,2,4] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[5,6,3,4,1,2] => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [5,3,2,6,1,4] => ? = 2
[6,2,1,5,3,4] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[6,2,4,1,3,5] => [5,1]
 => [[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ? = 0
[1,2,3,4,7,5,6] => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [5,4,3,2,1,6,7] => ? = 0
[1,2,5,3,7,4,6] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,2,7,3,4,5,6] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0
[1,2,7,3,6,4,5] => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [3,2,1,4,5,6,7] => ? = 0

Description

The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation

$\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety''

$X_\sigma$ . Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for

$\sigma$ . It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].

Matching statistic: St001816

Mapping

Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 33%

Values

[1,2] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[-1,2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[1,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 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
[-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
[-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
[1,2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[1,2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[1,-2,3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[-1,2,3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => ? = 0
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 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
[1,2,4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[1,2,-4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 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,-2,4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[1,-2,-4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-1,2,4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-1,2,-4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 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
[1,3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[1,3,-2,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[1,-3,2,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 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,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-1,3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-1,-3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[1,3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[1,3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 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,-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,-2,-3] => [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,-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
[1,4,3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[1,4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[1,-4,3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 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
[1,-4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-1,4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-1,-4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 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
[2,1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[2,1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[2,-1,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[-2,1,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 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,-1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-2,-1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[2,1,4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 0
[2,1,-4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 0
[-2,-1,4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 0
[-2,-1,-4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 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,-1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 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,-3,1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[2,4,3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[2,-4,3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[-2,4,3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 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,-2,4] => [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,-1,2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 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
[3,2,1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[3,2,-1,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[3,-2,1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-3,2,1,4] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 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,-1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-3,-2,-1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[3,2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[3,2,-4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[-3,2,4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[-3,2,-4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[3,4,1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 0
[3,-4,1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 0
[-3,4,-1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 0
[-3,-4,-1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => ? = 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,-1,3,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => ? = 0
[4,2,3,-1] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0
[4,2,-3,1] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[4,-2,3,1] => [2,1]
 => [1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 0
[-4,2,3,1] => [1,1]
 => [1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 0

Description

Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition

$\lambda$ has dimension equal to the number of standard tableaux of shape

$\lambda$ . Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape

$\lambda$ ; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].

Matching statistic: St000782
(load all 2 compositions to match this statistic)

Mapping

Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 33%

Values

[1,2] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[-1,2,3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,1,3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-2,-1,3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[3,2,1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-3,2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => ? = 0 + 1
[1,2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[1,2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,-2,3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => ? = 0 + 1
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,2,4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,2,-4,3] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,2,-4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,-2,4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,-2,-4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,2,4,3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,2,-4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,3,-2,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,-3,2,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,-3,-2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,-3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,-3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,-3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,-3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,-4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,-4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,4,3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[1,-4,3,2] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[1,-4,3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[1,-4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-1,-4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[2,1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,-1,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[-2,1,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[-2,-1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[-2,-1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-2,-1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[2,1,4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[2,1,-4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[-2,-1,4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[-2,-1,-4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[2,3,1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[2,-3,-1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-2,3,-1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-2,-3,1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[2,4,3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[2,-4,3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-2,4,3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-2,-4,3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[3,1,2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[3,-1,-2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-3,1,-2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-3,-1,2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[3,2,1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[3,2,-1,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[3,-2,1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-3,2,1,4] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[-3,2,-1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => ? = 0 + 1
[-3,2,-1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-3,-2,-1,4] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[3,2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[3,2,-4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-3,2,4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[-3,2,-4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[3,4,1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[3,-4,1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[-3,4,-1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[-3,-4,-1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => ? = 0 + 1
[4,1,3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[4,-1,3,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => ? = 0 + 1
[4,2,3,-1] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1
[4,2,-3,1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[4,-2,3,1] => [2,1]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1 = 0 + 1
[-4,2,3,1] => [1,1]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1 = 0 + 1

Description

The indicator function of whether a given perfect matching is an L & P matching. An L&P matching is built inductively as follows: starting with either a single edge, or a hairpin

$([1,3],[2,4])$ , insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges. The number of L&P matchings is (see [thm. 1, 2])

$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$

Matching statistic: St001722

Mapping

Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 33%

Values

[1,2] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,2,3] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[1,2,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,-2,3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,2,3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[2,1,3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-2,-1,3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[3,2,1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-3,2,-1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => ? = 0 + 1
[1,2,3,-4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[1,2,-3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[1,2,-3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,-2,3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[1,-2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,-2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 10111000 => ? = 0 + 1
[-1,2,3,-4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,2,-3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-1,-2,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,2,4,3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,2,4,-3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,2,-4,3] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,2,-4,-3] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,-2,4,3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,-2,-4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-1,2,4,3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-1,2,-4,-3] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,3,2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,3,2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,3,-2,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,-3,2,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,-3,-2,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,-3,-2,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-1,3,2,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-1,-3,-2,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,3,4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,3,-4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,-3,4,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,-3,-4,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,4,2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,4,-2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,-4,2,-3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,-4,-2,3] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[1,4,3,2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,4,3,-2] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,4,-3,2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[1,-4,3,2] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[1,-4,3,-2] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[1,-4,-3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-1,4,3,2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-1,-4,3,-2] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[2,1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[2,1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[2,1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[2,-1,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-2,1,3,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-2,-1,3,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[-2,-1,3,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-2,-1,-3,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[2,1,4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 0 + 1
[2,1,-4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 0 + 1
[-2,-1,4,3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 0 + 1
[-2,-1,-4,-3] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 0 + 1
[2,3,1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[2,-3,-1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[-2,3,-1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[-2,-3,1,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[2,4,3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[2,-4,3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[-2,4,3,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[-2,-4,3,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[3,1,2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[3,-1,-2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[-3,1,-2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[-3,-1,2,4] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[3,2,1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[3,2,1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[3,2,-1,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[3,-2,1,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-3,2,1,4] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[-3,2,-1,4] => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 10110100 => ? = 0 + 1
[-3,2,-1,-4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-3,-2,-1,4] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[3,2,4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[3,2,-4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[-3,2,4,-1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[-3,2,-4,1] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[3,4,1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 0 + 1
[3,-4,1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 0 + 1
[-3,4,-1,2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 0 + 1
[-3,-4,-1,-2] => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 11001100 => ? = 0 + 1
[4,1,3,2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[4,-1,3,-2] => [3,1]
 => [1,1,0,1,0,0,1,0]
 => 11010010 => ? = 0 + 1
[4,2,3,-1] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1
[4,2,-3,1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[4,-2,3,1] => [2,1]
 => [1,0,1,0,1,0]
 => 101010 => 1 = 0 + 1
[-4,2,3,1] => [1,1]
 => [1,0,1,1,0,0]
 => 101100 => 1 = 0 + 1

Description

The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence

$01$ , or a trailing

$0$ . A peak is a subsequence

$10$ or a trailing

$1$ . Let

$P$ be the lattice on binary words of length

$n$ , where the covering elements of a word are obtained by replacing a valley with a peak. An interval

$[w_1, w_2]$ in

$P$ is small if

$w_2$ is obtained from

$w_1$ by replacing some valleys with peaks. This statistic counts the number of chains

$w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word

$0110$ :

$0110 < 1011 < 1101 < 1110 < 1111$ and

$0110 < 1010 < 1101 < 1110 < 1111.$

The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000068The number of minimal elements in a poset. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001625The Möbius invariant of a lattice. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001927Sparre Andersen's number of positives of a signed permutation. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001208The 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

$K[x]/(x^n)$ . St001490The number of connected components of a skew partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001768The number of reduced words of a signed permutation. St001720The minimal length of a chain of small intervals in a lattice. St001624The breadth of a lattice. 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. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000941The number of characters of the symmetric group whose value on the partition is even. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St000635The number of strictly order preserving maps of a poset into itself.