- FindStat

Your data matches 74 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000381
(load all 40 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer composition.

Matching statistic: St000684
(load all 4 compositions to match this statistic)

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000684: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The global dimension of the LNakayama algebra associated to a Dyck path. An n-LNakayama algebra is a quiver algebra with a directed line as a connected quiver with

$n$ points for

$n \geq 2$ . Number those points from the left to the right by

$0,1,\ldots,n-1$ . The algebra is then uniquely determined by the dimension

$c_i$ of the projective indecomposable modules at point

$i$ . Such algebras are then uniquely determined by lists of the form

$[c_0,c_1,...,c_{n-1}]$ with the conditions:

$c_{n-1}=1$ and

$c_i -1 \leq c_{i+1}$ for all

$i$ . The number of such algebras is then the

$n-1$ -st Catalan number

$C_{n-1}$ . One can get also an interpretation with Dyck paths by associating the top boundary of the Auslander-Reiten quiver (which is a Dyck path) to those algebras. Example: [3,4,3,3,2,1] corresponds to the Dyck path [1,1,0,1,1,0,0,1,0,0]. Conjecture: that there is an explicit bijection between

$n$ -LNakayama algebras with global dimension bounded by

$m$ and Dyck paths with height at most

$m$ . Examples: * For

$m=2$ , the number of Dyck paths with global dimension at most

$m$ starts for

$n \geq 2$ with 1,2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192. * For

$m=3$ , the number of Dyck paths with global dimension at most

$m$ starts for

$n \geq 2$ with 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418.

Matching statistic: St000147

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St000982
(load all 6 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ 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] => 11 => 11 => 2
[2] => 10 => 01 => 1
[1,1,1] => 111 => 111 => 3
[1,2] => 110 => 011 => 2
[2,1] => 101 => 001 => 2
[3] => 100 => 101 => 1
[1,1,1,1] => 1111 => 1111 => 4
[1,1,2] => 1110 => 0111 => 3
[1,2,1] => 1101 => 0011 => 2
[1,3] => 1100 => 1011 => 2
[2,1,1] => 1011 => 0001 => 3
[2,2] => 1010 => 1001 => 2
[3,1] => 1001 => 1101 => 2
[4] => 1000 => 0101 => 1
[1,1,1,1,1] => 11111 => 11111 => 5
[1,1,1,2] => 11110 => 01111 => 4
[1,1,2,1] => 11101 => 00111 => 3
[1,1,3] => 11100 => 10111 => 3
[1,2,1,1] => 11011 => 00011 => 3
[1,2,2] => 11010 => 10011 => 2
[1,3,1] => 11001 => 11011 => 2
[1,4] => 11000 => 01011 => 2
[2,1,1,1] => 10111 => 00001 => 4
[2,1,2] => 10110 => 10001 => 3
[2,2,1] => 10101 => 11001 => 2
[2,3] => 10100 => 01001 => 2
[3,1,1] => 10011 => 11101 => 3
[3,2] => 10010 => 01101 => 2
[4,1] => 10001 => 00101 => 2
[5] => 10000 => 10101 => 1
[1,1,1,1,1,1] => 111111 => 111111 => 6
[1,1,1,1,2] => 111110 => 011111 => 5
[1,1,1,2,1] => 111101 => 001111 => 4
[1,1,1,3] => 111100 => 101111 => 4
[1,1,2,1,1] => 111011 => 000111 => 3
[1,1,2,2] => 111010 => 100111 => 3
[1,1,3,1] => 111001 => 110111 => 3
[1,1,4] => 111000 => 010111 => 3
[1,2,1,1,1] => 110111 => 000011 => 4
[1,2,1,2] => 110110 => 100011 => 3
[1,2,2,1] => 110101 => 110011 => 2
[1,2,3] => 110100 => 010011 => 2
[1,3,1,1] => 110011 => 111011 => 3
[1,3,2] => 110010 => 011011 => 2
[1,4,1] => 110001 => 001011 => 2
[1,5] => 110000 => 101011 => 2
[2,1,1,1,1] => 101111 => 000001 => 5
[2,1,1,2] => 101110 => 100001 => 4
[2,1,2,1] => 101101 => 110001 => 3

Description

The length of the longest constant subword.

Matching statistic: St000010

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000013
(load all 14 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000676

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St000734

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The last entry in the first row of a standard tableau.

Matching statistic: St000983
(load all 4 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St000983: 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] => 11 => 11 => 10 => 2
[2] => 10 => 01 => 00 => 1
[1,1,1] => 111 => 111 => 101 => 3
[1,2] => 110 => 011 => 001 => 2
[2,1] => 101 => 001 => 011 => 2
[3] => 100 => 101 => 111 => 1
[1,1,1,1] => 1111 => 1111 => 1010 => 4
[1,1,2] => 1110 => 0111 => 0010 => 3
[1,2,1] => 1101 => 0011 => 0110 => 2
[1,3] => 1100 => 1011 => 1110 => 2
[2,1,1] => 1011 => 0001 => 0100 => 3
[2,2] => 1010 => 1001 => 1100 => 2
[3,1] => 1001 => 1101 => 1000 => 2
[4] => 1000 => 0101 => 0000 => 1
[1,1,1,1,1] => 11111 => 11111 => 10101 => 5
[1,1,1,2] => 11110 => 01111 => 00101 => 4
[1,1,2,1] => 11101 => 00111 => 01101 => 3
[1,1,3] => 11100 => 10111 => 11101 => 3
[1,2,1,1] => 11011 => 00011 => 01001 => 3
[1,2,2] => 11010 => 10011 => 11001 => 2
[1,3,1] => 11001 => 11011 => 10001 => 2
[1,4] => 11000 => 01011 => 00001 => 2
[2,1,1,1] => 10111 => 00001 => 01011 => 4
[2,1,2] => 10110 => 10001 => 11011 => 3
[2,2,1] => 10101 => 11001 => 10011 => 2
[2,3] => 10100 => 01001 => 00011 => 2
[3,1,1] => 10011 => 11101 => 10111 => 3
[3,2] => 10010 => 01101 => 00111 => 2
[4,1] => 10001 => 00101 => 01111 => 2
[5] => 10000 => 10101 => 11111 => 1
[1,1,1,1,1,1] => 111111 => 111111 => 101010 => 6
[1,1,1,1,2] => 111110 => 011111 => 001010 => 5
[1,1,1,2,1] => 111101 => 001111 => 011010 => 4
[1,1,1,3] => 111100 => 101111 => 111010 => 4
[1,1,2,1,1] => 111011 => 000111 => 010010 => 3
[1,1,2,2] => 111010 => 100111 => 110010 => 3
[1,1,3,1] => 111001 => 110111 => 100010 => 3
[1,1,4] => 111000 => 010111 => 000010 => 3
[1,2,1,1,1] => 110111 => 000011 => 010110 => 4
[1,2,1,2] => 110110 => 100011 => 110110 => 3
[1,2,2,1] => 110101 => 110011 => 100110 => 2
[1,2,3] => 110100 => 010011 => 000110 => 2
[1,3,1,1] => 110011 => 111011 => 101110 => 3
[1,3,2] => 110010 => 011011 => 001110 => 2
[1,4,1] => 110001 => 001011 => 011110 => 2
[1,5] => 110000 => 101011 => 111110 => 2
[2,1,1,1,1] => 101111 => 000001 => 010100 => 5
[2,1,1,2] => 101110 => 100001 => 110100 => 4
[2,1,2,1] => 101101 => 110001 => 100100 => 3

Description

The length of the longest alternating subword. This is the length of the longest consecutive subword of the form

$010...$ or of the form

$101...$ .

Matching statistic: St000392
(load all 7 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => 0 => 0 = 1 - 1
[1,1] => [2] => 10 => 01 => 1 = 2 - 1
[2] => [1,1] => 11 => 00 => 0 = 1 - 1
[1,1,1] => [3] => 100 => 011 => 2 = 3 - 1
[1,2] => [1,2] => 110 => 001 => 1 = 2 - 1
[2,1] => [2,1] => 101 => 010 => 1 = 2 - 1
[3] => [1,1,1] => 111 => 000 => 0 = 1 - 1
[1,1,1,1] => [4] => 1000 => 0111 => 3 = 4 - 1
[1,1,2] => [1,3] => 1100 => 0011 => 2 = 3 - 1
[1,2,1] => [2,2] => 1010 => 0101 => 1 = 2 - 1
[1,3] => [1,1,2] => 1110 => 0001 => 1 = 2 - 1
[2,1,1] => [3,1] => 1001 => 0110 => 2 = 3 - 1
[2,2] => [1,2,1] => 1101 => 0010 => 1 = 2 - 1
[3,1] => [2,1,1] => 1011 => 0100 => 1 = 2 - 1
[4] => [1,1,1,1] => 1111 => 0000 => 0 = 1 - 1
[1,1,1,1,1] => [5] => 10000 => 01111 => 4 = 5 - 1
[1,1,1,2] => [1,4] => 11000 => 00111 => 3 = 4 - 1
[1,1,2,1] => [2,3] => 10100 => 01011 => 2 = 3 - 1
[1,1,3] => [1,1,3] => 11100 => 00011 => 2 = 3 - 1
[1,2,1,1] => [3,2] => 10010 => 01101 => 2 = 3 - 1
[1,2,2] => [1,2,2] => 11010 => 00101 => 1 = 2 - 1
[1,3,1] => [2,1,2] => 10110 => 01001 => 1 = 2 - 1
[1,4] => [1,1,1,2] => 11110 => 00001 => 1 = 2 - 1
[2,1,1,1] => [4,1] => 10001 => 01110 => 3 = 4 - 1
[2,1,2] => [1,3,1] => 11001 => 00110 => 2 = 3 - 1
[2,2,1] => [2,2,1] => 10101 => 01010 => 1 = 2 - 1
[2,3] => [1,1,2,1] => 11101 => 00010 => 1 = 2 - 1
[3,1,1] => [3,1,1] => 10011 => 01100 => 2 = 3 - 1
[3,2] => [1,2,1,1] => 11011 => 00100 => 1 = 2 - 1
[4,1] => [2,1,1,1] => 10111 => 01000 => 1 = 2 - 1
[5] => [1,1,1,1,1] => 11111 => 00000 => 0 = 1 - 1
[1,1,1,1,1,1] => [6] => 100000 => 011111 => 5 = 6 - 1
[1,1,1,1,2] => [1,5] => 110000 => 001111 => 4 = 5 - 1
[1,1,1,2,1] => [2,4] => 101000 => 010111 => 3 = 4 - 1
[1,1,1,3] => [1,1,4] => 111000 => 000111 => 3 = 4 - 1
[1,1,2,1,1] => [3,3] => 100100 => 011011 => 2 = 3 - 1
[1,1,2,2] => [1,2,3] => 110100 => 001011 => 2 = 3 - 1
[1,1,3,1] => [2,1,3] => 101100 => 010011 => 2 = 3 - 1
[1,1,4] => [1,1,1,3] => 111100 => 000011 => 2 = 3 - 1
[1,2,1,1,1] => [4,2] => 100010 => 011101 => 3 = 4 - 1
[1,2,1,2] => [1,3,2] => 110010 => 001101 => 2 = 3 - 1
[1,2,2,1] => [2,2,2] => 101010 => 010101 => 1 = 2 - 1
[1,2,3] => [1,1,2,2] => 111010 => 000101 => 1 = 2 - 1
[1,3,1,1] => [3,1,2] => 100110 => 011001 => 2 = 3 - 1
[1,3,2] => [1,2,1,2] => 110110 => 001001 => 1 = 2 - 1
[1,4,1] => [2,1,1,2] => 101110 => 010001 => 1 = 2 - 1
[1,5] => [1,1,1,1,2] => 111110 => 000001 => 1 = 2 - 1
[2,1,1,1,1] => [5,1] => 100001 => 011110 => 4 = 5 - 1
[2,1,1,2] => [1,4,1] => 110001 => 001110 => 3 = 4 - 1
[2,1,2,1] => [2,3,1] => 101001 => 010110 => 2 = 3 - 1

Description

The length of the longest run of ones in a binary word.

The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000439The position of the first down step of a Dyck path. St001933The largest multiplicity of a part in an integer partition. St001062The maximal size of a block of a set partition. St000503The maximal difference between two elements in a common block. St001058The breadth of the ordered tree. St000686The finitistic dominant dimension of a Dyck path. St000025The number of initial rises of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001674The number of vertices of the largest induced star graph in the graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St000171The degree of the graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001826The maximal number of leaves on a vertex of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000730The maximal arc length of a set partition. St001118The acyclic chromatic index of a graph. St000451The length of the longest pattern of the form k 1 2. St001090The number of pop-stack-sorts needed to sort a permutation. St000306The bounce count of a Dyck path. St000662The staircase size of the code of a permutation. St000308The height of the tree associated to a permutation. St000209Maximum difference of elements in cycles. St000141The maximum drop size of a permutation. St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000956The maximal displacement of a permutation. St000028The number of stack-sorts needed to sort a permutation. St001268The size of the largest ordinal summand in the poset. St000651The maximal size of a rise in a permutation. St001372The length of a longest cyclic run of ones of a binary word. St000628The balance of a binary word. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001652The length of a longest interval of consecutive numbers. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000299The number of nonisomorphic vertex-induced subtrees. St000328The maximum number of child nodes in a tree. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000094The depth of an ordered tree. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001117The game chromatic index of a graph. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St000652The maximal difference between successive positions of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001651The Frankl number of a lattice. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ .