- FindStat

Identifier

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000864: Permutations ⟶ ℤ

Values

[1] => [1] => [1,0] => [1] => 0

[1,2] => [2] => [1,1,0,0] => [2,1] => 1

[2,1] => [1,1] => [1,0,1,0] => [1,2] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 4

[1,2,3,5,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 3

[1,2,4,3,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[1,2,4,5,3] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 3

[1,2,5,3,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[1,2,5,4,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2

[1,3,2,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2

[1,3,2,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[1,3,4,2,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[1,3,4,5,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 3

[1,3,5,2,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[1,3,5,4,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2

[1,4,2,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2

[1,4,2,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[1,4,3,2,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1

[1,4,3,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[1,4,5,2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[1,4,5,3,2] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2

[1,5,2,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2

[1,5,2,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[1,5,3,2,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1

[1,5,3,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[1,5,4,2,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1

[1,5,4,3,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1

[2,1,3,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 2

[2,1,3,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1

[2,1,4,3,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0

[2,1,4,5,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1

[2,1,5,3,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0

[2,1,5,4,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0

[2,3,1,4,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2

[2,3,1,5,4] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[2,3,4,1,5] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[2,3,4,5,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 3

[2,3,5,1,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[2,3,5,4,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2

[2,4,1,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2

[2,4,1,5,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[2,4,3,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1

[2,4,3,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[2,4,5,1,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[2,4,5,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2

[2,5,1,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2

[2,5,1,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[2,5,3,1,4] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1

[2,5,3,4,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[2,5,4,1,3] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1

[2,5,4,3,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1

[3,1,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 2

[3,1,2,5,4] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1

[3,1,4,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0

[3,1,4,5,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1

[3,1,5,2,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0

[3,1,5,4,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0

[3,2,1,4,5] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 1

[3,2,1,5,4] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 0

[3,2,4,1,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0

[3,2,4,5,1] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 1

[3,2,5,1,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 0

[3,2,5,4,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 0

[3,4,1,2,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2

[3,4,1,5,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[3,4,2,1,5] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 1

[3,4,2,5,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

[3,4,5,1,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 2

[3,4,5,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2

[3,5,1,2,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 2

[3,5,1,4,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 1

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Description

The number of circled entries of the shifted recording tableau of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of circled entries in $Q$.

Map

to non-crossing permutation

Description

Sends a Dyck path $D$ with valley at positions $\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation $\pi$ having descents $\{i_1,\ldots,i_k\}$ and whose inverse has descents $\{j_1,\ldots,j_k\}$.
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to $n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

descent composition

Description

The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.