Identifier
-
Mp00080:
Set partitions
—to permutation⟶
Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St001745: Permutations ⟶ ℤ
Values
{{1}} => [1] => [1] => [1] => 0
{{1,2}} => [2,1] => [2,1] => [2,1] => 0
{{1},{2}} => [1,2] => [1,2] => [1,2] => 0
{{1,2,3}} => [2,3,1] => [3,1,2] => [3,2,1] => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => [2,1,3] => 1
{{1,3},{2}} => [3,2,1] => [3,2,1] => [2,3,1] => 0
{{1},{2,3}} => [1,3,2] => [1,3,2] => [1,3,2] => 0
{{1},{2},{3}} => [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}} => [2,3,4,1] => [4,1,2,3] => [4,3,2,1] => 0
{{1,2,3},{4}} => [2,3,1,4] => [3,1,2,4] => [3,2,1,4] => 1
{{1,2,4},{3}} => [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 0
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
{{1,3,4},{2}} => [3,2,4,1] => [4,2,1,3] => [2,4,3,1] => 0
{{1,3},{2,4}} => [3,4,1,2] => [3,4,1,2] => [3,1,4,2] => 1
{{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 1
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 1
{{1},{2,3,4}} => [1,3,4,2] => [1,4,2,3] => [1,4,3,2] => 0
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}} => [4,2,3,1] => [4,2,3,1] => [2,3,4,1] => 0
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 0
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1,2,3,4,5}} => [2,3,4,5,1] => [5,1,2,3,4] => [5,4,3,2,1] => 0
{{1,2,3,4},{5}} => [2,3,4,1,5] => [4,1,2,3,5] => [4,3,2,1,5] => 1
{{1,2,3,5},{4}} => [2,3,5,4,1] => [5,1,2,4,3] => [4,5,3,2,1] => 0
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,1,2,5,4] => [3,2,1,5,4] => 2
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,1,2,4,5] => [3,2,1,4,5] => 2
{{1,2,4,5},{3}} => [2,4,3,5,1] => [5,1,3,2,4] => [3,5,4,2,1] => 0
{{1,2,4},{3,5}} => [2,4,5,1,3] => [4,1,5,2,3] => [4,2,1,5,3] => 1
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 1
{{1,2,5},{3,4}} => [2,5,4,3,1] => [5,1,4,3,2] => [4,3,5,2,1] => 1
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,3,4] => [2,1,5,4,3] => 3
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 4
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [5,1,3,4,2] => [3,4,5,2,1] => 0
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,5,4,3] => [2,1,4,5,3] => 3
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 3
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3
{{1,3,4,5},{2}} => [3,2,4,5,1] => [5,2,1,3,4] => [2,5,4,3,1] => 0
{{1,3,4},{2,5}} => [3,5,4,1,2] => [4,5,1,3,2] => [4,3,1,5,2] => 1
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [4,2,1,3,5] => [2,4,3,1,5] => 1
{{1,3,5},{2,4}} => [3,4,5,2,1] => [5,4,1,2,3] => [4,2,5,3,1] => 1
{{1,3},{2,4,5}} => [3,4,1,5,2] => [3,5,1,2,4] => [3,1,5,4,2] => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 3
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [5,2,1,4,3] => [2,4,5,3,1] => 0
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [3,5,1,4,2] => [3,1,4,5,2] => 2
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,2,1,5,4] => [2,3,1,5,4] => 2
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,2,1,4,5] => [2,3,1,4,5] => 2
{{1,4,5},{2,3}} => [4,3,2,5,1] => [5,3,2,1,4] => [3,2,5,4,1] => 2
{{1,4},{2,3,5}} => [4,3,5,1,2] => [4,5,2,1,3] => [4,1,5,3,2] => 1
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,3,2,1,5] => [3,2,4,1,5] => 3
{{1,5},{2,3,4}} => [5,3,4,2,1] => [5,4,2,3,1] => [4,3,2,5,1] => 1
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,2,3,4] => [1,5,4,3,2] => 0
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,2,3,5] => [1,4,3,2,5] => 1
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [5,3,2,4,1] => [3,2,4,5,1] => 2
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,5,2,4,3] => [1,4,5,3,2] => 0
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [5,2,3,1,4] => [2,3,5,4,1] => 0
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [4,5,3,1,2] => [3,4,1,5,2] => 1
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [4,2,5,1,3] => [2,4,1,5,3] => 1
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [4,2,3,1,5] => [2,3,4,1,5] => 1
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [5,4,3,2,1] => [3,4,2,5,1] => 1
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,5,3,2,4] => [1,3,5,4,2] => 0
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,4,5,2,3] => [1,4,2,5,3] => 1
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => [1,3,4,2,5] => 1
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [5,2,4,3,1] => [2,4,3,5,1] => 1
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => [1,4,3,5,2] => 1
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,3,4] => [1,2,5,4,3] => 0
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [5,2,3,4,1] => [2,3,4,5,1] => 0
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,3,4,2] => [1,3,4,5,2] => 0
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,4,3] => [1,2,4,5,3] => 0
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
{{1,2,3,4,5,6}} => [2,3,4,5,6,1] => [6,1,2,3,4,5] => [6,5,4,3,2,1] => 0
{{1,2,3,4,5},{6}} => [2,3,4,5,1,6] => [5,1,2,3,4,6] => [5,4,3,2,1,6] => 1
{{1,2,3,4,6},{5}} => [2,3,4,6,5,1] => [6,1,2,3,5,4] => [5,6,4,3,2,1] => 0
{{1,2,3,4},{5,6}} => [2,3,4,1,6,5] => [4,1,2,3,6,5] => [4,3,2,1,6,5] => 2
{{1,2,3,4},{5},{6}} => [2,3,4,1,5,6] => [4,1,2,3,5,6] => [4,3,2,1,5,6] => 2
{{1,2,3,5,6},{4}} => [2,3,5,4,6,1] => [6,1,2,4,3,5] => [4,6,5,3,2,1] => 0
{{1,2,3,5},{4,6}} => [2,3,5,6,1,4] => [5,1,2,6,3,4] => [5,3,2,1,6,4] => 1
{{1,2,3,5},{4},{6}} => [2,3,5,4,1,6] => [5,1,2,4,3,6] => [4,5,3,2,1,6] => 1
{{1,2,3,6},{4,5}} => [2,3,6,5,4,1] => [6,1,2,5,4,3] => [5,4,6,3,2,1] => 1
{{1,2,3},{4,5,6}} => [2,3,1,5,6,4] => [3,1,2,6,4,5] => [3,2,1,6,5,4] => 3
{{1,2,3},{4,5},{6}} => [2,3,1,5,4,6] => [3,1,2,5,4,6] => [3,2,1,5,4,6] => 4
{{1,2,3,6},{4},{5}} => [2,3,6,4,5,1] => [6,1,2,4,5,3] => [4,5,6,3,2,1] => 0
{{1,2,3},{4,6},{5}} => [2,3,1,6,5,4] => [3,1,2,6,5,4] => [3,2,1,5,6,4] => 3
{{1,2,3},{4},{5,6}} => [2,3,1,4,6,5] => [3,1,2,4,6,5] => [3,2,1,4,6,5] => 3
{{1,2,3},{4},{5},{6}} => [2,3,1,4,5,6] => [3,1,2,4,5,6] => [3,2,1,4,5,6] => 3
{{1,2,4,5,6},{3}} => [2,4,3,5,6,1] => [6,1,3,2,4,5] => [3,6,5,4,2,1] => 0
{{1,2,4,5},{3,6}} => [2,4,6,5,1,3] => [5,1,6,2,4,3] => [5,4,2,1,6,3] => 1
{{1,2,4,5},{3},{6}} => [2,4,3,5,1,6] => [5,1,3,2,4,6] => [3,5,4,2,1,6] => 1
{{1,2,4,6},{3,5}} => [2,4,5,6,3,1] => [6,1,5,2,3,4] => [5,3,6,4,2,1] => 1
{{1,2,4},{3,5,6}} => [2,4,5,1,6,3] => [4,1,6,2,3,5] => [4,2,1,6,5,3] => 2
{{1,2,4},{3,5},{6}} => [2,4,5,1,3,6] => [4,1,5,2,3,6] => [4,2,1,5,3,6] => 3
{{1,2,4,6},{3},{5}} => [2,4,3,6,5,1] => [6,1,3,2,5,4] => [3,5,6,4,2,1] => 0
{{1,2,4},{3,6},{5}} => [2,4,6,1,5,3] => [4,1,6,2,5,3] => [4,2,1,5,6,3] => 2
{{1,2,4},{3},{5,6}} => [2,4,3,1,6,5] => [4,1,3,2,6,5] => [3,4,2,1,6,5] => 2
{{1,2,4},{3},{5},{6}} => [2,4,3,1,5,6] => [4,1,3,2,5,6] => [3,4,2,1,5,6] => 2
{{1,2,5,6},{3,4}} => [2,5,4,3,6,1] => [6,1,4,3,2,5] => [4,3,6,5,2,1] => 2
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Description
The number of occurrences of the arrow pattern 13 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 $x_1 < x_2 < \dots < x_k$.
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.
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
inverse
Description
Sends a permutation to its inverse.
Map
to permutation
Description
Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.
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