- FindStat

Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St000922
(load all 25 compositions to match this statistic)

Mapping

St000922: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 1
=> 1
=> 2
=> 1
=> 1
=> 2
=> 3
=> 2
=> 2
=> 2
=> 2
=> 2
=> 2
=> 3
=> 4
=> 3
=> 2
=> 2
=> 2
=> 3
=> 2
=> 3
=> 3
=> 2
=> 3
=> 2
=> 2
=> 2
=> 3
=> 4

Description

The minimal number such that all substrings of this length are unique.

Matching statistic: St000145

Mapping

Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences

$p(5n+4) \equiv 0 \pmod 5$ and

$p(7n+6) \equiv 0 \pmod 7.$

Matching statistic: St000295
(load all 2 compositions to match this statistic)

Mapping

Mp00224: Binary words —runsort⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
St000295: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> 0 => 0 => 0 = 1 - 1
=> 1 => 1 => 0 = 1 - 1
=> 00 => 00 => 1 = 2 - 1
=> 01 => 10 => 0 = 1 - 1
=> 01 => 10 => 0 = 1 - 1
=> 11 => 11 => 1 = 2 - 1
=> 000 => 000 => 2 = 3 - 1
=> 001 => 010 => 1 = 2 - 1
=> 001 => 010 => 1 = 2 - 1
=> 011 => 101 => 1 = 2 - 1
=> 001 => 010 => 1 = 2 - 1
=> 011 => 101 => 1 = 2 - 1
=> 011 => 101 => 1 = 2 - 1
=> 111 => 111 => 2 = 3 - 1
=> 0000 => 0000 => 3 = 4 - 1
=> 0001 => 0010 => 1 = 2 - 1
=> 0001 => 0010 => 1 = 2 - 1
=> 0011 => 0101 => 2 = 3 - 1
=> 0001 => 0010 => 1 = 2 - 1
=> 0101 => 1010 => 2 = 3 - 1
=> 0011 => 0101 => 2 = 3 - 1
=> 0111 => 1011 => 1 = 2 - 1
=> 0001 => 0010 => 1 = 2 - 1
=> 0011 => 0101 => 2 = 3 - 1
=> 0011 => 0101 => 2 = 3 - 1
=> 0111 => 1011 => 1 = 2 - 1
=> 0011 => 0101 => 2 = 3 - 1
=> 0111 => 1011 => 1 = 2 - 1
=> 0111 => 1011 => 1 = 2 - 1
=> 1111 => 1111 => 3 = 4 - 1

Description

The length of the border of a binary word. The border of a word is the longest word which is both a proper prefix and a proper suffix, including a possible empty border.

Matching statistic: St000384
(load all 2 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal part of the shifted composition of an integer partition. A partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ is shifted into a composition by adding

$i-1$ to the

$i$ -th part. The statistic is then

$\operatorname{max}_i\{ \lambda_i + i - 1 \}$ . See also [[St000380]].

Matching statistic: St000380
(load all 2 compositions to match this statistic)

Mapping

Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000380: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. Put differently, this is the smallest number

$n$ such that the partition fits into the triangular partition

$(n-1,n-2,\dots,1)$ .

Matching statistic: St000062

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000062: 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] => 2
01 => [1,1] => [1,0,1,0]
 => [2,1] => 1
10 => [1,1] => [1,0,1,0]
 => [2,1] => 1
11 => [2] => [1,1,0,0]
 => [1,2] => 2
000 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 3
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] => 2
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] => 2
110 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 2
111 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 3
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
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] => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 3
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] => 3
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4

Description

The length of the longest increasing subsequence of the permutation.

Matching statistic: St000308

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000308: 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] => 2
01 => [1,1] => [1,0,1,0]
 => [2,1] => 1
10 => [1,1] => [1,0,1,0]
 => [2,1] => 1
11 => [2] => [1,1,0,0]
 => [1,2] => 2
000 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 3
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] => 2
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] => 2
110 => [2,1] => [1,1,0,0,1,0]
 => [3,1,2] => 2
111 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 3
0000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
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] => 3
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,3,4,1] => 3
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => 3
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] => 3
1111 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4

Description

The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.

Matching statistic: St000982

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00278: Binary words —rowmotion⟶ Binary words
St000982: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

=> [1] => 1 => 1 => 1
=> [1] => 1 => 1 => 1
=> [2] => 10 => 01 => 1
=> [1,1] => 11 => 11 => 2
=> [1,1] => 11 => 11 => 2
=> [2] => 10 => 01 => 1
=> [3] => 100 => 001 => 2
=> [2,1] => 101 => 110 => 2
=> [1,1,1] => 111 => 111 => 3
=> [1,2] => 110 => 011 => 2
=> [1,2] => 110 => 011 => 2
=> [1,1,1] => 111 => 111 => 3
=> [2,1] => 101 => 110 => 2
=> [3] => 100 => 001 => 2
=> [4] => 1000 => 0001 => 3
=> [3,1] => 1001 => 0110 => 2
=> [2,1,1] => 1011 => 1101 => 2
=> [2,2] => 1010 => 1100 => 2
=> [1,1,2] => 1110 => 0111 => 3
=> [1,1,1,1] => 1111 => 1111 => 4
=> [1,2,1] => 1101 => 1110 => 3
=> [1,3] => 1100 => 0011 => 2
=> [1,3] => 1100 => 0011 => 2
=> [1,2,1] => 1101 => 1110 => 3
=> [1,1,1,1] => 1111 => 1111 => 4
=> [1,1,2] => 1110 => 0111 => 3
=> [2,2] => 1010 => 1100 => 2
=> [2,1,1] => 1011 => 1101 => 2
=> [3,1] => 1001 => 0110 => 2
=> [4] => 1000 => 0001 => 3

Description

The length of the longest constant subword.

Matching statistic: St001091

Mapping

Mp00262: Binary words —poset of factors⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.

Matching statistic: St000340

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000340: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns

$110$ and

$001$ .

The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000444The length of the maximal rise of a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001644The dimension of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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

$eA$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 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

$eA$ . St001060The distinguishing index of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St000454The largest eigenvalue of a graph if it is integral. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001330The hat guessing number 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

$K[x]/(x^n)$ . St000264The girth of a graph, which is not a tree. 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. St000455The second largest eigenvalue of a graph if it is integral.