- FindStat

Your data matches 11 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000993
(load all 3 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St000297

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00095: Integer partitions —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]
 => 110 => 2
([(0,1)],2)
 => [2]
 => 100 => 1
([],3)
 => [1,1,1]
 => 1110 => 3
([(1,2)],3)
 => [2,1]
 => 1010 => 1
([(0,2),(1,2)],3)
 => [2,2]
 => 1100 => 2
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 1
([],4)
 => [1,1,1,1]
 => 11110 => 4
([(2,3)],4)
 => [2,1,1]
 => 10110 => 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => 11010 => 2
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 11100 => 3
([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 11100 => 3
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 10100 => 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 111100 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 11000 => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 1
([],5)
 => [1,1,1,1,1]
 => 111110 => 5
([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 110110 => 2
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 111010 => 3
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 111100 => 4
([(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 2
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 111010 => 3
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 11100 => 3
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 111100 => 4
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => 101010 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 1111010 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 1111100 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => 110010 => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 110100 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 11111100 => 6
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 111000 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 111100 => 4
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 11000 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,2,2,2]
 => 1111100 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2,2]
 => 1011100 => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 111000 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 110100 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => 100100 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 101000 => 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,3,2,2]
 => 1101100 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [3,3,3,3]
 => 1111000 => 4
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => 110000 => 2

Description

The number of leading ones in a binary word.

Matching statistic: St000326

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],2)
 => [1,1]
 => 110 => 001 => 3 = 2 + 1
([(0,1)],2)
 => [2]
 => 100 => 011 => 2 = 1 + 1
([],3)
 => [1,1,1]
 => 1110 => 0001 => 4 = 3 + 1
([(1,2)],3)
 => [2,1]
 => 1010 => 0101 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => 1100 => 0011 => 3 = 2 + 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1000 => 0111 => 2 = 1 + 1
([],4)
 => [1,1,1,1]
 => 11110 => 00001 => 5 = 4 + 1
([(2,3)],4)
 => [2,1,1]
 => 10110 => 01001 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => 11010 => 00101 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 11100 => 00011 => 4 = 3 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 0011 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 11100 => 00011 => 4 = 3 + 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 10010 => 01101 => 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 10100 => 01011 => 2 = 1 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 111100 => 000011 => 5 = 4 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 11000 => 00111 => 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 10000 => 01111 => 2 = 1 + 1
([],5)
 => [1,1,1,1,1]
 => 111110 => 000001 => 6 = 5 + 1
([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 010001 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 110110 => 001001 => 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 111010 => 000101 => 4 = 3 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 111100 => 000011 => 5 = 4 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 00101 => 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 111010 => 000101 => 4 = 3 + 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 11100 => 00011 => 4 = 3 + 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 100110 => 011001 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => 111100 => 000011 => 5 = 4 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => 101010 => 010101 => 2 = 1 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 010011 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 1111010 => 0000101 => 5 = 4 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 1111100 => 0000011 => 6 = 5 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => 110010 => 001101 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 010011 => 2 = 1 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 110100 => 001011 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => 11111100 => 00000011 => 7 = 6 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 111000 => 000111 => 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => 111100 => 000011 => 5 = 4 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => 01011 => 2 = 1 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 010011 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 11000 => 00111 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,2,2,2]
 => 1111100 => 0000011 => 6 = 5 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2,2]
 => 1011100 => 0100011 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 111000 => 000111 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 110100 => 001011 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 100010 => 011101 => 2 = 1 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => 100100 => 011011 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 101000 => 010111 => 2 = 1 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,3,2,2]
 => 1101100 => 0010011 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [3,3,3,3]
 => 1111000 => 0000111 => 5 = 4 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => 110000 => 001111 => 3 = 2 + 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: St000363
(load all 3 compositions to match this statistic)

Mapping

Mp00111: Graphs —complement⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph

$G$ is a subset

$S$ of the vertices of

$G$ such that each edge of

$G$ contains at least one vertex of

$S$ . A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of

$G$ . This is also the number of independent sets of maximal cardinality of

$G$ .

Matching statistic: St001038

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

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

Description

The minimal height of a column in the parallelogram polyomino associated with the Dyck path.

Matching statistic: St001733

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of weak left to right maxima of a Dyck path. A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its left.

Matching statistic: St001107

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 86% ●values known / values provided: 95%●distinct values known / distinct values provided: 86%

Values

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

Description

The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.

Matching statistic: St000617

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000617: Dyck paths ⟶ ℤResult quality: 84% ●values known / values provided: 84%●distinct values known / distinct values provided: 86%

Values

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

Description

The number of global maxima of a Dyck path.

Matching statistic: St000733

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%

Values

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

Description

The row containing the largest entry of a standard tableau.

Matching statistic: St001803

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau

$T$ is the skew row-strict tableau obtained by gluing two copies of

$T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals

$\max_C\big(2\ell(T) - \ell(C)\big)$ , where

$\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals

$0$ , if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.

The following 1 statistic also match your data. Click on any of them to see the details.

St000640The rank of the largest boolean interval in a poset.