- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001786: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 5\ q + q^{2}$

$F_{4} = 15\ q + 4\ q^{2} + 4\ q^{3} + q^{4}$

$F_{5} = 54\ q + 15\ q^{2} + 18\ q^{3} + 16\ q^{4} + 12\ q^{6} + q^{8} + 4\ q^{9}$

$F_{6} = 235\ q + 60\ q^{2} + 76\ q^{3} + 76\ q^{4} + 48\ q^{5} + 56\ q^{6} + 22\ q^{8} + 26\ q^{9} + 48\ q^{10} + 26\ q^{12} + 13\ q^{16} + 24\ q^{18} + 6\ q^{24} + 4\ q^{27}$

$F_{7} = 1237\ q + 270\ q^{2} + 346\ q^{3} + 355\ q^{4} + 264\ q^{5} + 492\ q^{6} + 101\ q^{8} + 112\ q^{9} + 336\ q^{10} + 172\ q^{12} + 288\ q^{15} + 88\ q^{16} + 142\ q^{18} + 216\ q^{20} + 100\ q^{24} + 48\ q^{25} + 34\ q^{27} + 96\ q^{30} + 19\ q^{32} + 62\ q^{36} + 96\ q^{40} + 18\ q^{48} + 48\ q^{50} + 36\ q^{54} + 24\ q^{60} + 12\ q^{64} + 18\ q^{72} + 4\ q^{81} + 6\ q^{96}$

Description

The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order.
Alternatively, remark that the monomials of the polynomial

$\prod_{k=1}^n (z_1+\dots +z_k)$ are in bijection with Dyck paths, regarded as superdiagonal paths, with

$n$ east steps: the exponent of

$z_i$ is the number of north steps before the

$i$ -th east step, see [2]. Thus, this statistic records the coefficients of the monomials.
A formula for the coefficient of

$z_1^{a_1}\dots z_n^{a_n}$ is provided in [3]:

$c_{(a_1,\dots,a_n)} = \prod_{k=1}^{n-1} \frac{n-k+1 - \sum_{i=k+1}^n a_i}{a_k!}.$
This polynomial arises in a partial symmetrization process as follows, see [1]. For

$w\in\frak{S}_n$ , let

$w\cdot F(x_1,\dots,x_n)=F(x_{w(1)},\dots,x_{w(n)})$ . Furthermore, let

$G(\mathbf{x},\mathbf{z}) = \prod_{k=1}^n\frac{x_1z_1+x_2z_2+\cdots+x_kz_k}{x_k-x_{k+1}}.$
Then

$\sum_{w\in\frak{S}_{n+1}}w\cdot G = \prod_{k=1}^n (z_1+\dots +z_k)$ .

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.