- FindStat

Your data matches 266 different statistics following compositions of up to 3 maps.
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Matching statistic: St000382
(load all 20 compositions to match this statistic)

Mapping

Mp00038: Integer compositions —reverse⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The first part of an integer composition.

Matching statistic: St000326
(load all 19 compositions to match this statistic)

Mapping

Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2] => 10 => 10 => 1 = 0 + 1
[1,2] => 110 => 110 => 1 = 0 + 1
[3] => 100 => 010 => 2 = 1 + 1
[1,1,2] => 1110 => 1110 => 1 = 0 + 1
[1,3] => 1100 => 0110 => 2 = 1 + 1
[2,2] => 1010 => 1100 => 1 = 0 + 1
[4] => 1000 => 0010 => 3 = 2 + 1
[1,1,1,2] => 11110 => 11110 => 1 = 0 + 1
[1,1,3] => 11100 => 01110 => 2 = 1 + 1
[1,2,2] => 11010 => 11100 => 1 = 0 + 1
[1,4] => 11000 => 00110 => 3 = 2 + 1
[2,1,2] => 10110 => 11010 => 1 = 0 + 1
[2,3] => 10100 => 01100 => 2 = 1 + 1
[3,2] => 10010 => 10100 => 1 = 0 + 1
[5] => 10000 => 00010 => 4 = 3 + 1
[1,1,1,1,2] => 111110 => 111110 => 1 = 0 + 1
[1,1,1,3] => 111100 => 011110 => 2 = 1 + 1
[1,1,2,2] => 111010 => 111100 => 1 = 0 + 1
[1,1,4] => 111000 => 001110 => 3 = 2 + 1
[1,2,1,2] => 110110 => 111010 => 1 = 0 + 1
[1,2,3] => 110100 => 011100 => 2 = 1 + 1
[1,3,2] => 110010 => 101100 => 1 = 0 + 1
[1,5] => 110000 => 000110 => 4 = 3 + 1
[2,1,1,2] => 101110 => 110110 => 1 = 0 + 1
[2,1,3] => 101100 => 011010 => 2 = 1 + 1
[2,2,2] => 101010 => 111000 => 1 = 0 + 1
[2,4] => 101000 => 001100 => 3 = 2 + 1
[3,1,2] => 100110 => 101010 => 1 = 0 + 1
[3,3] => 100100 => 010100 => 2 = 1 + 1
[4,2] => 100010 => 100100 => 1 = 0 + 1
[6] => 100000 => 000010 => 5 = 4 + 1
[1,1,1,1,1,2] => 1111110 => 1111110 => 1 = 0 + 1
[1,1,1,1,3] => 1111100 => 0111110 => 2 = 1 + 1
[1,1,1,2,2] => 1111010 => 1111100 => 1 = 0 + 1
[1,1,1,4] => 1111000 => 0011110 => 3 = 2 + 1
[1,1,2,1,2] => 1110110 => 1111010 => 1 = 0 + 1
[1,1,2,3] => 1110100 => 0111100 => 2 = 1 + 1
[1,1,3,2] => 1110010 => 1011100 => 1 = 0 + 1
[1,1,5] => 1110000 => 0001110 => 4 = 3 + 1
[1,2,1,1,2] => 1101110 => 1110110 => 1 = 0 + 1
[1,2,1,3] => 1101100 => 0111010 => 2 = 1 + 1
[1,2,2,2] => 1101010 => 1111000 => 1 = 0 + 1
[1,2,4] => 1101000 => 0011100 => 3 = 2 + 1
[1,3,1,2] => 1100110 => 1011010 => 1 = 0 + 1
[1,3,3] => 1100100 => 0101100 => 2 = 1 + 1
[1,4,2] => 1100010 => 1001100 => 1 = 0 + 1
[1,6] => 1100000 => 0000110 => 5 = 4 + 1
[2,1,1,1,2] => 1011110 => 1101110 => 1 = 0 + 1
[2,1,1,3] => 1011100 => 0110110 => 2 = 1 + 1
[2,1,2,2] => 1011010 => 1110100 => 1 = 0 + 1

Description

The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of

$\{1,\dots,n,n+1\}$ that contains

$n+1$ , this is the minimal element of the set.

Matching statistic: St001176
(load all 3 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2] => ([],2)
 => [1,1]
 => 1 = 0 + 1
[1,2] => ([(1,2)],3)
 => [2,1]
 => 1 = 0 + 1
[3] => ([],3)
 => [1,1,1]
 => 2 = 1 + 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2 = 1 + 1
[2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 1 = 0 + 1
[4] => ([],4)
 => [1,1,1,1]
 => 3 = 2 + 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 1 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 3 = 2 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 2 = 1 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 1 = 0 + 1
[5] => ([],5)
 => [1,1,1,1,1]
 => 4 = 3 + 1
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 2 = 1 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 3 = 2 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 2 = 1 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 4 = 3 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 2 = 1 + 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => 3 = 2 + 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => 2 = 1 + 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 1 = 0 + 1
[6] => ([],6)
 => [1,1,1,1,1,1]
 => 5 = 4 + 1
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 2 = 1 + 1
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => 3 = 2 + 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 2 = 1 + 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => 4 = 3 + 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 2 = 1 + 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => 3 = 2 + 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 2 = 1 + 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1
[1,6] => ([(5,6)],7)
 => [2,1,1,1,1,1]
 => 5 = 4 + 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 2 = 1 + 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 1 = 0 + 1

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St000010
(load all 3 compositions to match this statistic)

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2] => ([],2)
 => [1,1]
 => 2 = 0 + 2
[1,2] => ([(1,2)],3)
 => [2,1]
 => 2 = 0 + 2
[3] => ([],3)
 => [1,1,1]
 => 3 = 1 + 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 2 = 0 + 2
[1,3] => ([(2,3)],4)
 => [2,1,1]
 => 3 = 1 + 2
[2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 2 = 0 + 2
[4] => ([],4)
 => [1,1,1,1]
 => 4 = 2 + 2
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 0 + 2
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 1 + 2
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 0 + 2
[1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 4 = 2 + 2
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 0 + 2
[2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3 = 1 + 2
[3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 2 = 0 + 2
[5] => ([],5)
 => [1,1,1,1,1]
 => 5 = 3 + 2
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 0 + 2
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 3 = 1 + 2
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 0 + 2
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => 4 = 2 + 2
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 0 + 2
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 3 = 1 + 2
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 0 + 2
[1,5] => ([(4,5)],6)
 => [2,1,1,1,1]
 => 5 = 3 + 2
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 0 + 2
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => 3 = 1 + 2
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 0 + 2
[2,4] => ([(3,5),(4,5)],6)
 => [3,1,1,1]
 => 4 = 2 + 2
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 0 + 2
[3,3] => ([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => 3 = 1 + 2
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => [5,1]
 => 2 = 0 + 2
[6] => ([],6)
 => [1,1,1,1,1,1]
 => 6 = 4 + 2
[1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2
[1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 3 = 1 + 2
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2
[1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => 4 = 2 + 2
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 3 = 1 + 2
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2
[1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => 5 = 3 + 2
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 3 = 1 + 2
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => 4 = 2 + 2
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 3 = 1 + 2
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2
[1,6] => ([(5,6)],7)
 => [2,1,1,1,1,1]
 => 6 = 4 + 2
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => 3 = 1 + 2
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [6,1]
 => 2 = 0 + 2

Description

The length of the partition.

Matching statistic: St000297
(load all 11 compositions to match this statistic)

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2] => [1,1] => 11 => 2 = 0 + 2
[1,2] => [1,2] => 110 => 2 = 0 + 2
[3] => [1,1,1] => 111 => 3 = 1 + 2
[1,1,2] => [1,3] => 1100 => 2 = 0 + 2
[1,3] => [1,1,2] => 1110 => 3 = 1 + 2
[2,2] => [1,2,1] => 1101 => 2 = 0 + 2
[4] => [1,1,1,1] => 1111 => 4 = 2 + 2
[1,1,1,2] => [1,4] => 11000 => 2 = 0 + 2
[1,1,3] => [1,1,3] => 11100 => 3 = 1 + 2
[1,2,2] => [1,2,2] => 11010 => 2 = 0 + 2
[1,4] => [1,1,1,2] => 11110 => 4 = 2 + 2
[2,1,2] => [1,3,1] => 11001 => 2 = 0 + 2
[2,3] => [1,1,2,1] => 11101 => 3 = 1 + 2
[3,2] => [1,2,1,1] => 11011 => 2 = 0 + 2
[5] => [1,1,1,1,1] => 11111 => 5 = 3 + 2
[1,1,1,1,2] => [1,5] => 110000 => 2 = 0 + 2
[1,1,1,3] => [1,1,4] => 111000 => 3 = 1 + 2
[1,1,2,2] => [1,2,3] => 110100 => 2 = 0 + 2
[1,1,4] => [1,1,1,3] => 111100 => 4 = 2 + 2
[1,2,1,2] => [1,3,2] => 110010 => 2 = 0 + 2
[1,2,3] => [1,1,2,2] => 111010 => 3 = 1 + 2
[1,3,2] => [1,2,1,2] => 110110 => 2 = 0 + 2
[1,5] => [1,1,1,1,2] => 111110 => 5 = 3 + 2
[2,1,1,2] => [1,4,1] => 110001 => 2 = 0 + 2
[2,1,3] => [1,1,3,1] => 111001 => 3 = 1 + 2
[2,2,2] => [1,2,2,1] => 110101 => 2 = 0 + 2
[2,4] => [1,1,1,2,1] => 111101 => 4 = 2 + 2
[3,1,2] => [1,3,1,1] => 110011 => 2 = 0 + 2
[3,3] => [1,1,2,1,1] => 111011 => 3 = 1 + 2
[4,2] => [1,2,1,1,1] => 110111 => 2 = 0 + 2
[6] => [1,1,1,1,1,1] => 111111 => 6 = 4 + 2
[1,1,1,1,1,2] => [1,6] => 1100000 => 2 = 0 + 2
[1,1,1,1,3] => [1,1,5] => 1110000 => 3 = 1 + 2
[1,1,1,2,2] => [1,2,4] => 1101000 => 2 = 0 + 2
[1,1,1,4] => [1,1,1,4] => 1111000 => 4 = 2 + 2
[1,1,2,1,2] => [1,3,3] => 1100100 => 2 = 0 + 2
[1,1,2,3] => [1,1,2,3] => 1110100 => 3 = 1 + 2
[1,1,3,2] => [1,2,1,3] => 1101100 => 2 = 0 + 2
[1,1,5] => [1,1,1,1,3] => 1111100 => 5 = 3 + 2
[1,2,1,1,2] => [1,4,2] => 1100010 => 2 = 0 + 2
[1,2,1,3] => [1,1,3,2] => 1110010 => 3 = 1 + 2
[1,2,2,2] => [1,2,2,2] => 1101010 => 2 = 0 + 2
[1,2,4] => [1,1,1,2,2] => 1111010 => 4 = 2 + 2
[1,3,1,2] => [1,3,1,2] => 1100110 => 2 = 0 + 2
[1,3,3] => [1,1,2,1,2] => 1110110 => 3 = 1 + 2
[1,4,2] => [1,2,1,1,2] => 1101110 => 2 = 0 + 2
[1,6] => [1,1,1,1,1,2] => 1111110 => 6 = 4 + 2
[2,1,1,1,2] => [1,5,1] => 1100001 => 2 = 0 + 2
[2,1,1,3] => [1,1,4,1] => 1110001 => 3 = 1 + 2
[2,1,2,2] => [1,2,3,1] => 1101001 => 2 = 0 + 2

Description

The number of leading ones in a binary word.

Matching statistic: St000377

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The dinv defect of an integer partition. This is the number of cells

$c$ in the diagram of an integer partition

$\lambda$ for which

$\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$ .

Matching statistic: St001091

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St001382

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of boxes in the diagram of a partition that do not lie in its Durfee square.

Matching statistic: St001384

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001384: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains.

Matching statistic: St001714

Mapping

Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001714: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. In particular, partitions with statistic

$0$ are wide partitions.

The following 256 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000148The number of odd parts of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000228The size of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000548The number of different non-empty partial sums of an integer partition. St000744The length of the path to the largest entry in a standard Young tableau. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001127The sum of the squares of the parts of a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001933The largest multiplicity of a part in an integer partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000708The product of the parts of an integer partition. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001279The sum of the parts of an integer partition that are at least two. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001733The number of weak left to right maxima of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St001172The number of 1-rises at odd height of a Dyck path. St000012The area of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000617The number of global maxima of a Dyck path. St000984The number of boxes below precisely one peak. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000918The 2-limited packing number of a graph. St000383The last part of an integer composition. St000439The position of the first down step of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000011The number of touch points (or returns) of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000993The multiplicity of the largest part of an integer partition. St000013The height of a Dyck path. St000674The number of hills of a Dyck path. St000759The smallest missing part in an integer partition. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000068The number of minimal elements in a poset. St000203The number of external nodes of a binary tree. St000069The number of maximal elements of a poset. St000007The number of saliances of the permutation. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000505The biggest entry in the block containing the 1. St000971The smallest closer of a set partition. St001050The number of terminal closers of a set partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St000504The cardinality of the first block of a set partition. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000823The number of unsplittable factors of the set partition. St000258The burning number of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001829The common independence number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000917The open packing number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001463The number of distinct columns in the nullspace of a graph. St001672The restrained domination number of a graph. St001691The number of kings in a graph. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000931The number of occurrences of the pattern UUU in a Dyck path. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000444The length of the maximal rise of a Dyck path. St000700The protection number of an ordered tree. St000722The number of different neighbourhoods in a graph. St000908The length of the shortest maximal antichain in a poset. St000909The number of maximal chains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St000287The number of connected components of a graph. St000553The number of blocks of a graph. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St000286The number of connected components of the complement of a graph. St000822The Hadwiger number of the graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001458The rank of the adjacency matrix of a graph. St001963The tree-depth of a graph. St000237The number of small exceedances. St000260The radius of a connected graph. St000654The first descent of a permutation. St001828The Euler characteristic of a graph. St001812The biclique partition number of a graph. St000740The last entry of a permutation. St001368The number of vertices of maximal degree in a graph. St001461The number of topologically connected components of the chord diagram of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000546The number of global descents of a permutation. St000054The first entry of the permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000883The number of longest increasing subsequences of a permutation. St000989The number of final rises of a permutation. St001330The hat guessing number of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000996The number of exclusive left-to-right maxima of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000843The decomposition number of a perfect matching. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000181The number of connected components of the Hasse diagram for the poset. St001462The number of factors of a standard tableaux under concatenation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000454The largest eigenvalue of a graph if it is integral. St000031The number of cycles in the cycle decomposition of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001662The length of the longest factor of consecutive numbers in a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000090The variation of a composition. St000335The difference of lower and upper interactions. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St000314The number of left-to-right-maxima of a permutation. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call 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 special CNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000217The number of occurrences of the pattern 312 in a permutation. St000315The number of isolated vertices of a graph. St000338The number of pixed points of a permutation. St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000051The size of the left subtree of a binary tree. St000056The decomposition (or block) number of a permutation. St000133The "bounce" of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000456The monochromatic index of a connected graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001192The maximal dimension of

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

$S$ over the corresponding Nakayama algebra

$A$ . St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001592The maximal number of simple paths between any two different vertices of a graph. St000015The number of peaks of a Dyck path. St000061The number of nodes on the left branch of a binary tree. St000084The number of subtrees. St000144The pyramid weight of the Dyck path. St000328The maximum number of child nodes in a tree. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000374The number of exclusive right-to-left minima of a permutation. St000308The height of the tree associated to a permutation. St001644The dimension of a graph. St001060The distinguishing index of a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001557The number of inversions of the second entry of a permutation. St000732The number of double deficiencies 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. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001530The depth of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000028The number of stack-sorts needed to sort a permutation. St001651The Frankl number of a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001866The nesting alignments of a signed permutation. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice.