- FindStat

Your data matches 85 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000630
(load all 11 compositions to match this statistic)

Mapping

St000630: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 1
=> 1
=> 1
=> 2
=> 2
=> 1
=> 1
=> 2
=> 1
=> 2
=> 2
=> 1
=> 2
=> 1
=> 1
=> 2
=> 2
=> 2
=> 2
=> 2
=> 1
=> 2
=> 2
=> 1
=> 2
=> 2
=> 2
=> 2
=> 2
=> 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: St001673
(load all 3 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> [2] => 0 = 1 - 1
=> [1,1] => 0 = 1 - 1
=> [3] => 0 = 1 - 1
=> [2,1] => 1 = 2 - 1
=> [1,2] => 1 = 2 - 1
=> [1,1,1] => 0 = 1 - 1
=> [4] => 0 = 1 - 1
=> [3,1] => 1 = 2 - 1
=> [2,2] => 0 = 1 - 1
=> [2,1,1] => 1 = 2 - 1
=> [1,3] => 1 = 2 - 1
=> [1,2,1] => 0 = 1 - 1
=> [1,1,2] => 1 = 2 - 1
=> [1,1,1,1] => 0 = 1 - 1
=> [5] => 0 = 1 - 1
=> [4,1] => 1 = 2 - 1
=> [3,2] => 1 = 2 - 1
=> [3,1,1] => 1 = 2 - 1
=> [2,3] => 1 = 2 - 1
=> [2,2,1] => 1 = 2 - 1
=> [2,1,2] => 0 = 1 - 1
=> [2,1,1,1] => 1 = 2 - 1
=> [1,4] => 1 = 2 - 1
=> [1,3,1] => 0 = 1 - 1
=> [1,2,2] => 1 = 2 - 1
=> [1,2,1,1] => 1 = 2 - 1
=> [1,1,3] => 1 = 2 - 1
=> [1,1,2,1] => 1 = 2 - 1
=> [1,1,1,2] => 1 = 2 - 1
=> [1,1,1,1,1] => 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: St000007

Mapping

Mp00097: Binary words —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

0 => [1] => [1,0]
 => [1] => 1
1 => [1] => [1,0]
 => [1] => 1
00 => [2] => [1,1,0,0]
 => [1,2] => 1
01 => [1,1] => [1,0,1,0]
 => [2,1] => 2
10 => [1,1] => [1,0,1,0]
 => [2,1] => 2
11 => [2] => [1,1,0,0]
 => [1,2] => 1
000 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 1
001 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 2
010 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1,3] => 1
011 => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 2
100 => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1,3] => 1
110 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 2
111 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 2
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 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 2 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000402: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [2] => [1,1,0,0]
 => [2,1] => 1
1 => [1,1] => [1,0,1,0]
 => [1,2] => 1
00 => [3] => [1,1,1,0,0,0]
 => [3,2,1] => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [2,1,3] => 2
10 => [1,2] => [1,0,1,1,0,0]
 => [1,3,2] => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,2,3] => 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [4,3,2,1,5] => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [3,2,1,5,4] => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [3,2,1,4,5] => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,4,3] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,5,4,3,2] => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,4,3,2,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,4,3] => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => 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: St000526

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000526: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [2] => [[2],[]]
 => ([(0,1)],2)
 => 1
1 => [1,1] => [[1,1],[]]
 => ([(0,1)],2)
 => 1
00 => [3] => [[3],[]]
 => ([(0,2),(2,1)],3)
 => 1
01 => [2,1] => [[2,2],[1]]
 => ([(0,2),(1,2)],3)
 => 2
10 => [1,2] => [[2,1],[]]
 => ([(0,1),(0,2)],3)
 => 2
11 => [1,1,1] => [[1,1,1],[]]
 => ([(0,2),(2,1)],3)
 => 1
000 => [4] => [[4],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
001 => [3,1] => [[3,3],[2]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
010 => [2,2] => [[3,2],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
100 => [1,3] => [[3,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
101 => [1,2,1] => [[2,2,1],[1]]
 => ([(0,3),(1,2),(1,3)],4)
 => 1
110 => [1,1,2] => [[2,1,1],[]]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
0000 => [5] => [[5],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
0001 => [4,1] => [[4,4],[3]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
0010 => [3,2] => [[4,3],[2]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => 2
0100 => [2,3] => [[4,2],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 2
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => 2
1000 => [1,4] => [[4,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
1001 => [1,3,1] => [[3,3,1],[2]]
 => ([(0,4),(1,2),(1,3),(3,4)],5)
 => 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => ([(0,3),(0,4),(1,2),(1,4)],5)
 => 2
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => ([(0,3),(1,2),(1,4),(3,4)],5)
 => 2
1100 => [1,1,3] => [[3,1,1],[]]
 => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => ([(0,4),(1,2),(1,4),(2,3)],5)
 => 2
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 2
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1

Description

The number of posets with combinatorially isomorphic order polytopes.

Matching statistic: St000903

Mapping

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 1 => 1 => [1,1] => 1
=> 1 => 1 => [1,1] => 1
=> 01 => 00 => [3] => 1
=> 10 => 11 => [1,1,1] => 1
=> 11 => 10 => [1,2] => 2
=> 11 => 10 => [1,2] => 2
=> 001 => 011 => [2,1,1] => 2
=> 010 => 000 => [4] => 1
=> 101 => 111 => [1,1,1,1] => 1
=> 101 => 111 => [1,1,1,1] => 1
=> 101 => 111 => [1,1,1,1] => 1
=> 110 => 100 => [1,3] => 2
=> 111 => 101 => [1,2,1] => 2
=> 111 => 101 => [1,2,1] => 2
=> 0001 => 0100 => [2,3] => 2
=> 0010 => 0111 => [2,1,1,1] => 2
=> 0101 => 0000 => [5] => 1
=> 0101 => 0000 => [5] => 1
=> 1001 => 1100 => [1,1,3] => 2
=> 1010 => 1111 => [1,1,1,1,1] => 1
=> 1011 => 1110 => [1,1,1,2] => 2
=> 1011 => 1110 => [1,1,1,2] => 2
=> 1001 => 1100 => [1,1,3] => 2
=> 1010 => 1111 => [1,1,1,1,1] => 1
=> 1101 => 1000 => [1,4] => 2
=> 1101 => 1000 => [1,4] => 2
=> 1101 => 1000 => [1,4] => 2
=> 1110 => 1011 => [1,2,1,1] => 2
=> 1111 => 1010 => [1,2,2] => 2
=> 1111 => 1010 => [1,2,2] => 2

Description

The number of different parts of an integer composition.

Matching statistic: St001043

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [2] => [1,1,0,0]
 => [(1,4),(2,3)]
 => 1
1 => [1,1] => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
00 => [3] => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
10 => [1,2] => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 2
000 => [4] => [1,1,1,1,0,0,0,0]
 => [(1,8),(2,7),(3,6),(4,5)]
 => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [(1,4),(2,3),(5,8),(6,7)]
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [(1,2),(3,8),(4,7),(5,6)]
 => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [(1,2),(3,4),(5,8),(6,7)]
 => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 2
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [(1,10),(2,9),(3,8),(4,7),(5,6)]
 => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [(1,8),(2,7),(3,6),(4,5),(9,10)]
 => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [(1,6),(2,5),(3,4),(7,10),(8,9)]
 => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8),(9,10)]
 => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [(1,4),(2,3),(5,10),(6,9),(7,8)]
 => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [(1,4),(2,3),(5,8),(6,7),(9,10)]
 => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [(1,4),(2,3),(5,6),(7,10),(8,9)]
 => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8),(9,10)]
 => 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [(1,2),(3,10),(4,9),(5,8),(6,7)]
 => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [(1,2),(3,8),(4,7),(5,6),(9,10)]
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [(1,2),(3,6),(4,5),(7,10),(8,9)]
 => 2
1011 => [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
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [(1,2),(3,4),(5,10),(6,9),(7,8)]
 => 1
1101 => [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
1110 => [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
1111 => [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

Description

The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. The bijection between perfect matchings of

$\{1,\dots,2n\}$ and trees with

$n+1$ leaves is described in Example 5.2.6 of [1].

Matching statistic: St001359
(load all 2 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => [2] => [1,1,0,0]
 => [1,2] => 1
1 => [1,1] => [1,0,1,0]
 => [2,1] => 1
00 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 1
01 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 2
10 => [1,2] => [1,0,1,1,0,0]
 => [2,3,1] => 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => [2,1,3] => 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [3,1,4,2] => 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,4,1,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [3,1,4,5,2] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [3,1,4,2,5] => 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,5,1,3,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,5] => 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => 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: St001884

Mapping

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St001884: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 1 => 0 => 0 => 1
=> 1 => 0 => 0 => 1
=> 01 => 11 => 11 => 2
=> 10 => 00 => 00 => 2
=> 11 => 01 => 10 => 1
=> 11 => 01 => 10 => 1
=> 001 => 101 => 011 => 1
=> 010 => 110 => 101 => 2
=> 101 => 001 => 010 => 2
=> 101 => 001 => 010 => 2
=> 101 => 001 => 010 => 2
=> 110 => 010 => 100 => 1
=> 111 => 011 => 110 => 1
=> 111 => 011 => 110 => 1
=> 0001 => 1001 => 0011 => 1
=> 0010 => 1010 => 0101 => 2
=> 0101 => 1101 => 1011 => 2
=> 0101 => 1101 => 1011 => 2
=> 1001 => 0001 => 0010 => 2
=> 1010 => 0010 => 0100 => 2
=> 1011 => 0011 => 0110 => 2
=> 1011 => 0011 => 0110 => 2
=> 1001 => 0001 => 0010 => 2
=> 1010 => 0010 => 0100 => 2
=> 1101 => 0101 => 1010 => 2
=> 1101 => 0101 => 1010 => 2
=> 1101 => 0101 => 1010 => 2
=> 1110 => 0110 => 1100 => 1
=> 1111 => 0111 => 1110 => 1
=> 1111 => 0111 => 1110 => 1

Description

The number of borders of a binary word. A border of a binary word

$w$ is a word which is both a prefix and a suffix of

$w$ .

Matching statistic: St000143

Mapping

Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => 1 => [1] => [1]
 => 0 = 1 - 1
1 => 1 => [1] => [1]
 => 0 = 1 - 1
00 => 01 => [1,1] => [1,1]
 => 1 = 2 - 1
01 => 10 => [1,1] => [1,1]
 => 1 = 2 - 1
10 => 11 => [2] => [2]
 => 0 = 1 - 1
11 => 11 => [2] => [2]
 => 0 = 1 - 1
000 => 001 => [2,1] => [2,1]
 => 0 = 1 - 1
001 => 010 => [1,1,1] => [1,1,1]
 => 1 = 2 - 1
010 => 101 => [1,1,1] => [1,1,1]
 => 1 = 2 - 1
011 => 101 => [1,1,1] => [1,1,1]
 => 1 = 2 - 1
100 => 101 => [1,1,1] => [1,1,1]
 => 1 = 2 - 1
101 => 110 => [2,1] => [2,1]
 => 0 = 1 - 1
110 => 111 => [3] => [3]
 => 0 = 1 - 1
111 => 111 => [3] => [3]
 => 0 = 1 - 1
0000 => 0001 => [3,1] => [3,1]
 => 0 = 1 - 1
0001 => 0010 => [2,1,1] => [2,1,1]
 => 1 = 2 - 1
0010 => 0101 => [1,1,1,1] => [1,1,1,1]
 => 1 = 2 - 1
0011 => 0101 => [1,1,1,1] => [1,1,1,1]
 => 1 = 2 - 1
0100 => 1001 => [1,2,1] => [2,1,1]
 => 1 = 2 - 1
0101 => 1010 => [1,1,1,1] => [1,1,1,1]
 => 1 = 2 - 1
0110 => 1011 => [1,1,2] => [2,1,1]
 => 1 = 2 - 1
0111 => 1011 => [1,1,2] => [2,1,1]
 => 1 = 2 - 1
1000 => 1001 => [1,2,1] => [2,1,1]
 => 1 = 2 - 1
1001 => 1010 => [1,1,1,1] => [1,1,1,1]
 => 1 = 2 - 1
1010 => 1101 => [2,1,1] => [2,1,1]
 => 1 = 2 - 1
1011 => 1101 => [2,1,1] => [2,1,1]
 => 1 = 2 - 1
1100 => 1101 => [2,1,1] => [2,1,1]
 => 1 = 2 - 1
1101 => 1110 => [3,1] => [3,1]
 => 0 = 1 - 1
1110 => 1111 => [4] => [4]
 => 0 = 1 - 1
1111 => 1111 => [4] => [4]
 => 0 = 1 - 1

Description

The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.

The following 75 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000480The number of lower covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000628The balance of a binary word. St000665The number of rafts of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001280The number of parts of an integer partition that are at least two. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St000259The diameter of a connected graph. St000260The radius of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001399The distinguishing number of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000544The cop number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St000379The number of Hamiltonian cycles in a graph. St001071The beta invariant of the graph. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001570The minimal number of edges to add to make a graph Hamiltonian. St000454The largest eigenvalue of a graph if it is integral. St001624The breadth of a lattice. St001644The dimension of a graph. St001330The hat guessing number of a graph. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001877Number of indecomposable injective modules with projective dimension 2. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000741The Colin de Verdière graph invariant. St001118The acyclic chromatic index of a graph. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. 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. St000668The least common multiple of the parts of the partition. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.