- FindStat

Identifier

Mp00240: Permutations —weak exceedance partition⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001744: Permutations ⟶ ℤ

Values

[1] => {{1}} => [1] => [1] => 0

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

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

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

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

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

[2,3,1] => {{1,2,3}} => [2,3,1] => [3,1,2] => 1

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

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

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

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

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

[1,3,4,2] => {{1},{2,3,4}} => [1,3,4,2] => [1,4,2,3] => 1

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

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

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

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

[2,3,1,4] => {{1,2,3},{4}} => [2,3,1,4] => [3,1,2,4] => 1

[2,3,4,1] => {{1,2,3,4}} => [2,3,4,1] => [4,1,2,3] => 2

[2,4,1,3] => {{1,2,4},{3}} => [2,4,3,1] => [3,4,1,2] => 1

[2,4,3,1] => {{1,2,4},{3}} => [2,4,3,1] => [3,4,1,2] => 1

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

[3,1,4,2] => {{1,3,4},{2}} => [3,2,4,1] => [2,4,1,3] => 1

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

[3,2,4,1] => {{1,3,4},{2}} => [3,2,4,1] => [2,4,1,3] => 1

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

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

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

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

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

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

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

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

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

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

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

[1,2,4,5,3] => {{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,3,4] => 1

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

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

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

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

[1,3,4,2,5] => {{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,2,3,5] => 1

[1,3,4,5,2] => {{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,2,3,4] => 2

[1,3,5,2,4] => {{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,4,5,2,3] => 1

[1,3,5,4,2] => {{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,4,5,2,3] => 1

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

[1,4,2,5,3] => {{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,3,5,2,4] => 1

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

[1,4,3,5,2] => {{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,3,5,2,4] => 1

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

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

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

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

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

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

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

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

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

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

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

[2,1,4,5,3] => {{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,3,4] => 1

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

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

[2,3,1,4,5] => {{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,1,2,4,5] => 1

[2,3,1,5,4] => {{1,2,3},{4,5}} => [2,3,1,5,4] => [3,1,2,5,4] => 1

[2,3,4,1,5] => {{1,2,3,4},{5}} => [2,3,4,1,5] => [4,1,2,3,5] => 2

[2,3,4,5,1] => {{1,2,3,4,5}} => [2,3,4,5,1] => [5,1,2,3,4] => 3

[2,3,5,1,4] => {{1,2,3,5},{4}} => [2,3,5,4,1] => [4,5,1,2,3] => 2

[2,3,5,4,1] => {{1,2,3,5},{4}} => [2,3,5,4,1] => [4,5,1,2,3] => 2

[2,4,1,3,5] => {{1,2,4},{3},{5}} => [2,4,3,1,5] => [3,4,1,2,5] => 1

[2,4,1,5,3] => {{1,2,4,5},{3}} => [2,4,3,5,1] => [3,5,1,2,4] => 2

[2,4,3,1,5] => {{1,2,4},{3},{5}} => [2,4,3,1,5] => [3,4,1,2,5] => 1

[2,4,3,5,1] => {{1,2,4,5},{3}} => [2,4,3,5,1] => [3,5,1,2,4] => 2

[2,4,5,1,3] => {{1,2,4},{3,5}} => [2,4,5,1,3] => [4,1,2,5,3] => 1

[2,4,5,3,1] => {{1,2,4},{3,5}} => [2,4,5,1,3] => [4,1,2,5,3] => 1

[2,5,1,3,4] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,4,5,1,2] => 1

[2,5,1,4,3] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,4,5,1,2] => 1

[2,5,3,1,4] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,4,5,1,2] => 1

[2,5,3,4,1] => {{1,2,5},{3},{4}} => [2,5,3,4,1] => [3,4,5,1,2] => 1

[2,5,4,1,3] => {{1,2,5},{3,4}} => [2,5,4,3,1] => [4,3,5,1,2] => 1

[2,5,4,3,1] => {{1,2,5},{3,4}} => [2,5,4,3,1] => [4,3,5,1,2] => 1

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

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

[3,1,4,2,5] => {{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,4,1,3,5] => 1

[3,1,4,5,2] => {{1,3,4,5},{2}} => [3,2,4,5,1] => [2,5,1,3,4] => 2

[3,1,5,2,4] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,4,5,1,3] => 1

[3,1,5,4,2] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,4,5,1,3] => 1

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

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

[3,2,4,1,5] => {{1,3,4},{2},{5}} => [3,2,4,1,5] => [2,4,1,3,5] => 1

[3,2,4,5,1] => {{1,3,4,5},{2}} => [3,2,4,5,1] => [2,5,1,3,4] => 2

[3,2,5,1,4] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,4,5,1,3] => 1

[3,2,5,4,1] => {{1,3,5},{2},{4}} => [3,2,5,4,1] => [2,4,5,1,3] => 1

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

[3,4,1,5,2] => {{1,3},{2,4,5}} => [3,4,1,5,2] => [3,1,5,2,4] => 1

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

[3,4,2,5,1] => {{1,3},{2,4,5}} => [3,4,1,5,2] => [3,1,5,2,4] => 1

[3,4,5,1,2] => {{1,3,5},{2,4}} => [3,4,5,2,1] => [4,2,5,1,3] => 1

[3,4,5,2,1] => {{1,3,5},{2,4}} => [3,4,5,2,1] => [4,2,5,1,3] => 1

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

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

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Description

The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation.
Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$
such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$.
Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation.
An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows.
Let $\Phi$ be the map Mp00087inverse first fundamental transformation. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.

Map

inverse first fundamental transformation

Description

Let $\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps $\sigma$ to the permutation $[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

weak exceedance partition

Description

The set partition induced by the weak exceedances of a permutation.
This is the coarsest set partition that contains all arcs $(i, \pi(i))$ with $i\leq\pi(i)$.