- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000156: Permutations ⟶ ℤ

Values

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

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

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

[1,2,3] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 6

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

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

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

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

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

[1,2,3,4] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 10

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

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

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

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

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

[2,1,3,4] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [3,1,4,5,2] => 8

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => 15

[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,1,0,0,0] => [2,3,4,6,1,5] => 10

[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,1,0,0,1,0,0] => [2,3,5,1,6,4] => 11

[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,1,0,0,0] => [2,3,5,6,1,4] => 10

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

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

[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => [2,4,1,5,6,3] => 12

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

[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,1,6,3] => 11

[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,0,1,0,1,0,0,0] => [2,4,5,6,1,3] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,1,4,5,6,2] => 13

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

[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => [3,1,5,2,6,4] => 9

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

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

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

[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => [3,4,1,5,6,2] => 12

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

[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,0,1,0,0,1,0,0] => [3,4,5,1,6,2] => 11

[2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,4,5,6,1,2] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => 10

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

[3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [4,1,5,2,6,3] => 9

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

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

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

[3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,1,2,5,6,3] => 10

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

[3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => [4,1,5,2,6,3] => 9

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 154 entries. <<<

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searching the database for the individual values of this statistic

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searching the database for statistics with the same generating function

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$F_{1} = q$

$F_{2} = q + q^{3}$

$F_{3} = 2\ q + 2\ q^{3} + q^{4} + q^{6}$

$F_{4} = 6\ q + 6\ q^{3} + 3\ q^{4} + 2\ q^{5} + 3\ q^{6} + 2\ q^{7} + q^{8} + q^{10}$

$F_{5} = 24\ q + 24\ q^{3} + 12\ q^{4} + 8\ q^{5} + 18\ q^{6} + 8\ q^{7} + 10\ q^{8} + 3\ q^{9} + 6\ q^{10} + 3\ q^{11} + 2\ q^{12} + q^{13} + q^{15}$

Description

The Denert index of a permutation.
It is defined as

$\begin{align*} den(\sigma) &= \#\{ 1\leq l < k \leq n : \sigma(k) < \sigma(l) \leq k \} \\ &+ \#\{ 1\leq l < k \leq n : \sigma(l) \leq k < \sigma(k) \} \\ &+ \#\{ 1\leq l < k \leq n : k < \sigma(k) < \sigma(l) \} \end{align*}$
where

$n$ is the size of

$\sigma$ . It was studied by Denert in [1], and it was shown by Foata and Zeilberger in [2] that the bistatistic

$(exc,den)$ is Euler-Mahonian. Here,

$exc$ is the number of weak exceedences, see St000155The number of exceedances (also excedences) of a permutation..

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final down step.

Map

to 321-avoiding permutation (Krattenthaler)

Description

Krattenthaler's bijection to 321-avoiding permutations.
Draw the path of semilength

$n$ in an

$n\times n$ square matrix, starting at the upper left corner, with right and down steps, and staying below the diagonal. Then the permutation matrix is obtained by placing ones into the cells corresponding to the peaks of the path and placing ones into the remaining columns from left to right, such that the row indices of the cells increase.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.