- FindStat

Your data matches 59 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001673
(load all 4 compositions to match this statistic)

Mapping

St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The degree of asymmetry of an integer composition. This is the number of pairs of symmetrically positioned distinct entries.

Matching statistic: St000630
(load all 12 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => 1 => 1 => 1
[1,1] => 11 => 11 => 1
[2] => 10 => 00 => 1
[1,1,1] => 111 => 111 => 1
[1,2] => 110 => 001 => 2
[2,1] => 101 => 100 => 2
[3] => 100 => 010 => 1
[1,1,1,1] => 1111 => 1111 => 1
[1,1,2] => 1110 => 0011 => 2
[1,2,1] => 1101 => 1001 => 1
[1,3] => 1100 => 0101 => 2
[2,1,1] => 1011 => 1100 => 2
[2,2] => 1010 => 0000 => 1
[3,1] => 1001 => 1010 => 2
[4] => 1000 => 0110 => 1
[1,1,1,1,1] => 11111 => 11111 => 1
[1,1,1,2] => 11110 => 00111 => 2
[1,1,2,1] => 11101 => 10011 => 2
[1,1,3] => 11100 => 01011 => 2
[1,2,1,1] => 11011 => 11001 => 2
[1,2,2] => 11010 => 00001 => 2
[1,3,1] => 11001 => 10101 => 1
[1,4] => 11000 => 01101 => 2
[2,1,1,1] => 10111 => 11100 => 2
[2,1,2] => 10110 => 00100 => 1
[2,2,1] => 10101 => 10000 => 2
[2,3] => 10100 => 01000 => 2
[3,1,1] => 10011 => 11010 => 2
[3,2] => 10010 => 00010 => 2
[4,1] => 10001 => 10110 => 2
[5] => 10000 => 01110 => 1
[1,1,1,1,1,1] => 111111 => 111111 => 1
[1,1,2,1,1] => 111011 => 110011 => 1
[1,2,1,1,1] => 110111 => 111001 => 2
[1,2,2,1] => 110101 => 100001 => 1

Description

The length of the shortest palindromic decomposition of a binary word. A palindromic decomposition (paldec for short) of a word

$w=a_1,\dots,a_n$ is any list of factors

$p_1,\dots,p_k$ such that

$w=p_1\dots p_k$ and each

$p_i$ is a palindrome, i.e. coincides with itself read backwards.

Matching statistic: St001359
(load all 20 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. In other words, this is

$2^k$ where

$k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition. The generating function for the number of equivalence classes,

$f(n)$ , is

$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$

Matching statistic: St000007

Mapping

Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of saliances of the permutation. A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

Matching statistic: St000402
(load all 13 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
St000402: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [(1,2)]
 => [2,1] => 1
[1,1] => [1,0,1,0]
 => [(1,2),(3,4)]
 => [2,1,4,3] => 1
[2] => [1,1,0,0]
 => [(1,4),(2,3)]
 => [4,3,2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => 1
[1,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => 2
[2,1] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => 2
[3] => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => [2,1,4,3,6,5,8,7] => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => [2,1,4,3,8,7,6,5] => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => [2,1,6,5,4,3,8,7] => 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => [2,1,8,7,6,5,4,3] => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => [4,3,2,1,6,5,8,7] => 2
[2,2] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => [4,3,2,1,8,7,6,5] => 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => [6,5,4,3,2,1,8,7] => 2
[4] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => [8,7,6,5,4,3,2,1] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10)]
 => [2,1,4,3,6,5,8,7,10,9] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,6),(7,10),(8,9)]
 => [2,1,4,3,6,5,10,9,8,7] => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10)]
 => [2,1,4,3,8,7,6,5,10,9] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => [2,1,4,3,10,9,8,7,6,5] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10)]
 => [2,1,6,5,4,3,8,7,10,9] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => [2,1,6,5,4,3,10,9,8,7] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => [2,1,8,7,6,5,4,3,10,9] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => [2,1,10,9,8,7,6,5,4,3] => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => [4,3,2,1,6,5,8,7,10,9] => 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => [4,3,2,1,6,5,10,9,8,7] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => [4,3,2,1,8,7,6,5,10,9] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => [4,3,2,1,10,9,8,7,6,5] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => [6,5,4,3,2,1,8,7,10,9] => 2
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => [6,5,4,3,2,1,10,9,8,7] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => [8,7,6,5,4,3,2,1,10,9] => 2
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => [10,9,8,7,6,5,4,3,2,1] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
 => [2,1,4,3,6,5,8,7,10,9,12,11] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
 => [2,1,4,3,8,7,6,5,10,9,12,11] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8),(9,10),(11,12)]
 => [2,1,6,5,4,3,8,7,10,9,12,11] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9),(11,12)]
 => [2,1,6,5,4,3,10,9,8,7,12,11] => 1

Description

Half the size of the symmetry class of a permutation. The symmetry class of a permutation

$\pi$ is the set of all permutations that can be obtained from

$\pi$ by the three elementary operations '''inverse''' ([[Mp00066]]), '''reverse''' ([[Mp00064]]), and '''complement''' ([[Mp00069]]). This statistic is undefined for the unique permutation on one element, because its value would be

$1/2$ .

Matching statistic: St000486
(load all 20 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1,0]
 => [2,1] => [2,1] => 0 = 1 - 1
[1,1] => [1,0,1,0]
 => [3,1,2] => [1,3,2] => 0 = 1 - 1
[2] => [1,1,0,0]
 => [2,3,1] => [2,1,3] => 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[3] => [1,1,1,0,0,0]
 => [2,3,4,1] => [2,1,3,4] => 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [4,1,2,5,3] => 1 = 2 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,3,2,5,4] => 0 = 1 - 1
[1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,1,4,5,2] => 1 = 2 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [2,3,5,1,4] => 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [2,1,4,3,5] => 0 = 1 - 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [2,5,1,3,4] => 1 = 2 - 1
[4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,1,3,4,5] => 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [1,2,3,4,6,5] => 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [5,1,2,3,6,4] => 1 = 2 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [1,4,2,3,6,5] => 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [4,1,2,5,6,3] => 1 = 2 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [1,3,4,2,6,5] => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,1,5,2,6,4] => 1 = 2 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,4,1,2,6,5] => 0 = 1 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,4,5,6,2] => 1 = 2 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [2,3,4,6,1,5] => 1 = 2 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [2,1,5,6,3,4] => 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [2,4,1,6,3,5] => 1 = 2 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [2,1,4,5,3,6] => 1 = 2 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => [2,3,6,1,4,5] => 1 = 2 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => [2,1,5,3,4,6] => 1 = 2 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => [2,6,1,3,4,5] => 1 = 2 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => [2,1,3,4,5,6] => 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [7,1,2,3,4,5,6] => [1,2,3,4,5,7,6] => 0 = 1 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [4,1,2,7,3,5,6] => [1,2,4,3,5,7,6] => 0 = 1 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [3,1,7,2,4,5,6] => [1,3,4,5,2,7,6] => 1 = 2 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [3,1,5,2,7,4,6] => [1,3,2,5,4,7,6] => 0 = 1 - 1

Description

The number of cycles of length at least 3 of a permutation.

Matching statistic: St000506

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000506: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of standard desarrangement tableaux of shape equal to the given partition. A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry

$i$ such that

$i+1$ appears to the right or above

$i$ in the tableau (with respect to English tableau notation). This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also: * [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition * [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.

Matching statistic: St001440

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001440: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition.

Matching statistic: St000526
(load all 2 compositions to match this statistic)

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000526: Posets ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of posets with combinatorially isomorphic order polytopes.

Matching statistic: St000988

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000988: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%

Values

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

Description

The orbit size of a permutation under Foata's bijection.

The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000836The number of descents of distance 2 of a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000260The radius of a connected graph. St001624The breadth of a lattice. St001569The maximal modular displacement of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001768The number of reduced words of a signed permutation. St000089The absolute variation of a composition. St000456The monochromatic index of a connected graph. St000650The number of 3-rises of a permutation. St001822The number of alignments of a signed permutation. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001208The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000102The charge of a semistandard tableau. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001556The number of inversions of the third entry of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.