Descents

The number of descents of $p$ : positions $i$ with $p_i > p_{i+1}$ .

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaPermutationPermutationMathWorldPermutationWikidataQ161519Encyclopedia of MathematicsPermutationnLabpermutationBritannicatopic/permutationFindStatSt000291via BinaryWord✓St001946via ParkingFunction✓St000021via Permutation✓St001427via SignedPermutation✓St000157via StandardTableau✓

Descents(p)the descent count of a one-line permutation

p

SageMathenumeratio_stat("Permutation", p, "Descents")

Typed signatures

Type	Package	Applies
`(composition \| list \| permutation \| signed_permutation \| standard_tableau) -> integer \| number`	`combinatorics`	always

Domain: Permutation statistics

Details

The permutations of $\{1, \dots, n\}$ with $k$ descents are counted by the Eulerian number $\left\langle{n\atop k}\right\rangle$
Complementary to Ascents: every adjacent pair is one or the other, so $\mathrm{Descents} + \mathrm{Ascents} = n - 1$
Defined over Permutation as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
Takes a Permutation, and also a bare list of integers: this reading compares entries with each other rather than with their positions, so it stands on any sequence.

Definition

Epsil, as the implementation runs it.

on Permutation, over _x

0 if Length(_x) < 2 else Count(Filter(
                                       Range(1, Length(_x) - 1),
                                       Function(_x[i] > _x[i + 1], i),
                                     ))

Examples

Basic #

only $3 > 1$

In

\mathrm{Descents}(\bigl\lbrack3, 1, 2\bigr\rbrack)

Out

1

every step falls

In

\mathrm{Descents}(\bigl\lbrack3, 2, 1\bigr\rbrack)

Out

2

the one-line word $312$

In

\mathrm{Descents}({\operatorname{Permutation}}([3, 1, 2]))

Out

1

the one-line word $312$ , as a plain list

In

\mathrm{Descents}(\bigl\lbrack3, 1, 2\bigr\rbrack)

Out

1

the one-line word $2413$

In

\mathrm{Descents}({\operatorname{Permutation}}([2, 4, 1, 3]))

Out

1

the composition $[2,1,3,1,1]$

In

\mathrm{CombinatorialStat}({\operatorname{IntegerComposition}}([2, 1, 3, 1, 1]), \text{Descents})

Out

2

the signed permutation $\bar{2}\,3\,\bar{1}$ : $0 > -2$ and $3 > -1$

In

\mathrm{CombinatorialStat}({\operatorname{SignedPermutation}}([-2, 3, -1]), \text{Descents})

Out

2

the standard tableau with rows $13 \mid 24$ : $2$ is below $1$ , and $4$ is below $3$

In

\mathrm{CombinatorialStat}({\operatorname{StandardTableau}}([[1, 3], [2, 4]]), \text{Descents})

Out

2

the word $1332$

In

\mathrm{CombinatorialStat}({\operatorname{Word}}([1, 3, 3, 2]), \text{Descents})

Out

1

the binary word $10100$

In

\mathrm{CombinatorialStat}({\operatorname{BinaryWord}}([1, 0, 1, 0, 0]), \text{Descents})

Out

2

the arrangement $314$

In

\mathrm{CombinatorialStat}({\operatorname{Arrangement}}([3, 1, 4]), \text{Descents})

Out

1

the parking function $211$

In

\mathrm{CombinatorialStat}({\operatorname{ParkingFunction}}([2, 1, 1]), \text{Descents})

Out

1

the surjection $2121$

In

\mathrm{CombinatorialStat}({\operatorname{Surjection}}([2, 1, 2, 1]), \text{Descents})

Out

2

Properties #

descents and ascents partition the $n - 1$ steps

In

\mathrm{Descents}(\bigl\lbrack3, 1, 2\bigr\rbrack)+\mathrm{Ascents}(\bigl\lbrack3, 1, 2\bigr\rbrack)=2

Out

\mathrm{True}

FindStat's value for [1]

In

\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1]), \text{Descents})

Out

0

Known value · FindStat St000021

FindStat's value for [1,2]

In

\mathrm{CombinatorialStat}({\operatorname{Permutation}}([1, 2]), \text{Descents})

Out

0

Known value · FindStat St000021

FindStat's value for [2,1]

In

\mathrm{CombinatorialStat}({\operatorname{Permutation}}([2, 1]), \text{Descents})

Out

1

Known value · FindStat St000021

About

Examples	17
Checked against	SageMath 8 agree
Crosswalk	Wikipedia ×2 MathWorld Wikidata Encyclopedia of Mathematics nLab Britannica FindStat ×5SageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

See also: Ascents, MajorIndex, SymmetricGroup