- FindStat

Identifier

Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000647: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation Mp00090cycle-as-one-line notation, see [Theorem 2.5, 1].
For the number of small descents, see St000214The number of adjacencies of a permutation..

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

invert Laguerre heap

Description

The permutation obtained by inverting the corresponding Laguerre heap, according to Viennot.
Let $\pi$ be a permutation. Following Viennot [1], we associate to $\pi$ a heap of pieces, by considering each decreasing run $(\pi_i, \pi_{i+1}, \dots, \pi_j)$ of $\pi$ as one piece, beginning with the left most run. Two pieces commute if and only if the minimal element of one piece is larger than the maximal element of the other piece.
This map yields the permutation corresponding to the heap obtained by reversing the reading direction of the heap.
Equivalently, this is the permutation obtained by flipping the noncrossing arc diagram of Reading [2] vertically.
By definition, this map preserves the set of decreasing runs.