- FindStat

Your data matches 23 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000993
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The multiplicity of the largest part of an integer partition.

Matching statistic: St000326

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => 1110 => 0111 => 2 = 1 + 1
([(0,2),(1,2)],3)
 => [2,2]
 => 1100 => 0011 => 3 = 2 + 1
([],4)
 => [1,1,1,1]
 => 11110 => 01111 => 2 = 1 + 1
([(2,3)],4)
 => [2,1,1]
 => 10110 => 01101 => 2 = 1 + 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => 11010 => 01011 => 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 11100 => 00111 => 3 = 2 + 1
([(0,3),(1,2)],4)
 => [2,2]
 => 1100 => 0011 => 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 11100 => 00111 => 3 = 2 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 10100 => 00101 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 111100 => 001111 => 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 11000 => 00011 => 4 = 3 + 1
([],5)
 => [1,1,1,1,1]
 => 111110 => 011111 => 2 = 1 + 1
([(3,4)],5)
 => [2,1,1,1]
 => 101110 => 011101 => 2 = 1 + 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => 110110 => 011011 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => 111010 => 010111 => 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => 111100 => 001111 => 3 = 2 + 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => 11010 => 01011 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => 111010 => 010111 => 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => 11100 => 00111 => 3 = 2 + 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 => 001111 => 3 = 2 + 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 => 001101 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => 1111010 => 0101111 => 2 = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => 1111100 => 0011111 => 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => 110010 => 010011 => 2 = 1 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 001101 => 3 = 2 + 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 => 00111111 => 3 = 2 + 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 => 001111 => 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => 10100 => 00101 => 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => 101100 => 001101 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 11000 => 00011 => 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,2,2,2]
 => 1111100 => 0011111 => 3 = 2 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2,2]
 => 1011100 => 0011101 => 3 = 2 + 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
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => 100100 => 001001 => 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 101000 => 000101 => 4 = 3 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,3,2,2]
 => 1101100 => 0011011 => 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 => 0001111 => 4 = 3 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => 110000 => 000011 => 5 = 4 + 1
([],6)
 => [1,1,1,1,1,1]
 => 1111110 => 0111111 => 2 = 1 + 1
([(4,5)],6)
 => [2,1,1,1,1]
 => 1011110 => 0111101 => 2 = 1 + 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => 1101110 => 0111011 => 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => 1110110 => 0110111 => 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [2,2,2,2,1]
 => 1111010 => 0101111 => 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [2,2,2,2,2]
 => 1111100 => 0011111 => 3 = 2 + 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => 110110 => 011011 => 2 = 1 + 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,3,2,2,2,2,1]
 => 1101111010 => 0101111011 => ? = 1 + 1
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,3,2,2,2,2,1]
 => 1101111010 => 0101111011 => ? = 1 + 1
([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [4,2,2,2,2,2]
 => 1001111100 => 0011111001 => ? = 2 + 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,3,2,2,2,2]
 => 1010111100 => 0011110101 => ? = 2 + 1
([(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)
 => [4,4,2,2,2,2]
 => 1100111100 => 0011110011 => ? = 2 + 1
([(0,2),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,3,3,2,2,2]
 => 1011011100 => 0011101101 => ? = 2 + 1
([(0,3),(0,6),(1,3),(1,5),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
 => [4,2,2,2,2,2]
 => 1001111100 => 0011111001 => ? = 2 + 1
([(0,3),(0,6),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [4,3,2,2,2,2]
 => 1010111100 => 0011110101 => ? = 2 + 1
([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
 => [4,3,3,2,2,2]
 => 1011011100 => 0011101101 => ? = 2 + 1
([(0,2),(0,6),(1,2),(1,4),(1,5),(2,3),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,3,2,2,2,2]
 => 1010111100 => 0011110101 => ? = 2 + 1
([(0,1),(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [4,4,2,2,2,2]
 => 1100111100 => 0011110011 => ? = 2 + 1
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => [4,3,2,2,2,2]
 => 1010111100 => 0011110101 => ? = 2 + 1
([(0,1),(0,2),(0,5),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,3,3,2,2,2]
 => 1011011100 => 0011101101 => ? = 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: St001322
(load all 3 compositions to match this statistic)

Mapping

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St001322: Graphs ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The size of a minimal independent dominating set in a graph.

Matching statistic: St000297

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [3]
 => 1000 => 1
([(0,2),(1,2)],3)
 => [2,2]
 => [2,2]
 => 1100 => 2
([],4)
 => [1,1,1,1]
 => [4]
 => 10000 => 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 10010 => 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => [3,2]
 => 10100 => 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => 11000 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 1100 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => 11000 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2,2,1]
 => 11010 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [4,4]
 => 110000 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [2,2,2]
 => 11100 => 3
([],5)
 => [1,1,1,1,1]
 => [5]
 => 100000 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 100010 => 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [4,2]
 => 100100 => 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [4,3]
 => 101000 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => [4,4]
 => 110000 => 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 10100 => 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [4,3]
 => 101000 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [3,3]
 => 11000 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 100110 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => [4,4]
 => 110000 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [3,2,1]
 => 101010 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [3,3,1]
 => 110010 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => [5,4]
 => 1010000 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => [5,5]
 => 1100000 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [3,2,2]
 => 101100 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [3,3,1]
 => 110010 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [3,3,2]
 => 110100 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,2,2]
 => [6,6]
 => 11000000 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => [3,3,3]
 => 111000 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => [4,4]
 => 110000 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 11010 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [3,3,1]
 => 110010 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => 11100 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,2,2,2]
 => [5,5]
 => 1100000 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2,2]
 => [4,4,1]
 => 1100010 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => [3,3,3]
 => 111000 => 3
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [3,3,2]
 => 110100 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [2,2,1,1]
 => 110110 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [2,2,2,1]
 => 111010 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,3,2,2]
 => [4,4,2]
 => 1100100 => 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]
 => 1110000 => 3
([(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]
 => 111100 => 4
([],6)
 => [1,1,1,1,1,1]
 => [6]
 => 1000000 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [5,1]
 => 1000010 => 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [5,2]
 => 1000100 => 1
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => [5,3]
 => 1001000 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [2,2,2,2,1]
 => [5,4]
 => 1010000 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [2,2,2,2,2]
 => [5,5]
 => 1100000 => 2
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [4,2]
 => 100100 => 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [2,2,2,2,2,2,1,1]
 => [8,6]
 => 1001000000 => ? = 1
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [3,2,2,2,2,2,1]
 => [7,6,1]
 => 1010000010 => ? = 1
([(0,6),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,3,2,2,2,2,1]
 => [7,6,2]
 => 1010000100 => ? = 1
([(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,3,2,2,2,2,1]
 => [7,6,2]
 => 1010000100 => ? = 1
([(0,6),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,5),(1,4),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,4),(0,6),(1,4),(1,6),(2,5),(2,6),(3,4),(3,5),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,5),(1,3),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,1),(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,1),(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,3),(0,6),(1,2),(1,6),(2,4),(3,5),(4,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => [3,2,2,2,2,2,1]
 => [7,6,1]
 => 1010000010 => ? = 1
([(0,6),(1,4),(1,5),(2,3),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,3),(0,4),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,4),(0,6),(1,4),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [4,2,2,2,2,2]
 => [6,6,1,1]
 => 1100000110 => ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [4,3,2,2,2,2]
 => [6,6,2,1]
 => 1100001010 => ? = 2
([(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)
 => [4,4,2,2,2,2]
 => [6,6,2,2]
 => 1100001100 => ? = 2
([(0,3),(0,6),(1,3),(1,5),(2,4),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)
 => [4,2,2,2,2,2]
 => [6,6,1,1]
 => 1100000110 => ? = 2
([(0,3),(0,6),(1,3),(1,4),(1,6),(2,4),(2,5),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [4,3,2,2,2,2]
 => [6,6,2,1]
 => 1100001010 => ? = 2
([(0,4),(0,6),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5)],7)
 => [3,2,2,2,2,2,2]
 => [7,7,1]
 => 1100000010 => ? = 2
([(0,2),(0,6),(1,2),(1,4),(1,5),(2,3),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [4,3,2,2,2,2]
 => [6,6,2,1]
 => 1100001010 => ? = 2
([(0,1),(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [4,4,2,2,2,2]
 => [6,6,2,2]
 => 1100001100 => ? = 2
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
 => [4,3,2,2,2,2]
 => [6,6,2,1]
 => 1100001010 => ? = 2

Description

The number of leading ones in a binary word.

Matching statistic: St001038
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 81% ●values known / values provided: 81%●distinct values known / distinct values provided: 100%

Values

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

Description

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

Matching statistic: St001107

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([],4)
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 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]
 => 0 = 1 - 1
([(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]
 => 1 = 2 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
([(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]
 => 1 = 2 - 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 = 2 - 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]
 => 1 = 2 - 1
([(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 = 3 - 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]
 => 0 = 1 - 1
([(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]
 => 0 = 1 - 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]
 => 0 = 1 - 1
([(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]
 => 0 = 1 - 1
([(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]
 => 1 = 2 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
([(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]
 => 0 = 1 - 1
([(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]
 => 1 = 2 - 1
([(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]
 => 0 = 1 - 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]
 => 1 = 2 - 1
([(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]
 => 0 = 1 - 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 = 2 - 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]
 => 0 = 1 - 1
([(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]
 => 1 = 2 - 1
([(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]
 => 0 = 1 - 1
([(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 = 2 - 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]
 => 1 = 2 - 1
([(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]
 => 1 = 2 - 1
([(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]
 => 2 = 3 - 1
([(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]
 => 1 = 2 - 1
([(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 = 2 - 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 = 2 - 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 = 3 - 1
([(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]
 => 1 = 2 - 1
([(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 = 2 - 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]
 => 2 = 3 - 1
([(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]
 => 1 = 2 - 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 = 2 - 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]
 => 2 = 3 - 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]
 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [3,3,3,3]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => 2 = 3 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3 = 4 - 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [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]
 => 0 = 1 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [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]
 => 1 = 2 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,1]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [3,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2,2]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
 => [3,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(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]
 => ? = 1 - 1
([(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]
 => ? = 1 - 1
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => [2,2,2,2,2,2,1]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [2,2,2,2,2,2,1,1]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? = 1 - 1
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [2,2,2,2,2,2,1]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(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]
 => ? = 1 - 1
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => [2,2,2,2,2,2,1]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(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
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => [2,2,2,2,2,2,1]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 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
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => [2,2,2,2,2,2,1]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 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 - 1
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
 => [3,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [3,2,2,2,2,2,1]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [3,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,3,2,2,2,1]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,6),(1,4),(1,6),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
 => [3,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,3,2,2,2,2]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [2,2,2,2,2,2,2]
 => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,5),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,1),(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)
 => [3,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,3,2,2,2,1]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,3,2,2,2,2]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6)],7)
 => [3,2,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,3,3,2,2,2]
 => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
 => ? = 2 - 1
([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,3,2,2,2,2,1]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,1,0,0]
 => ?
 => ? = 1 - 1
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => [3,3,2,2,2,2]
 => [1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 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]
 => ? = 1 - 1
([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => [3,3,2,2,2,1]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,1,0,0,1,0]
 => ? = 1 - 1
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
 => [3,2,2,2,2,2]
 => [1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 2 - 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: St000733
(load all 2 compositions to match this statistic)

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 1
([(0,2),(1,2)],3)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([],4)
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 3
([],5)
 => [1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [2,2,2,2,2]
 => [5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [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]
 => [6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => [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]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,2,2,2]
 => [5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2,2]
 => [4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => [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]
 => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 3
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,3,2,2]
 => [4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8]]
 => 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]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
 => ? = 3
([(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]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 4
([],6)
 => [1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [2,2,2,2,1]
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => [2,2,2,2,2]
 => [5,5]
 => [[1,2,3,4,5],[6,7,8,9,10]]
 => 2
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => [2,2,2,2,2,1]
 => [6,5]
 => [[1,2,3,4,5,11],[6,7,8,9,10]]
 => ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,1]
 => [7,6]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2]
 => [7,7]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3]
 => [4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
 => ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [2,2,2,2,2,1]
 => [6,5]
 => [[1,2,3,4,5,11],[6,7,8,9,10]]
 => ? = 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,2,2,2]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? = 2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
 => [3,2,2,2,2]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? = 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2]
 => [7,7]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
 => ? = 2
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [3,2,2,2,2,2]
 => [6,6,1]
 => [[1,3,4,5,6,7],[2,9,10,11,12,13],[8]]
 => ? = 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,2,2,2]
 => [5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3]
 => [4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
 => ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2]
 => [5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => [6,6,2]
 => [[1,2,5,6,7,8],[3,4,11,12,13,14],[9,10]]
 => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [2,2,2,2,2,2,2]
 => [7,7]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
 => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,2,2,2]
 => [5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ? = 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,1]
 => [5,4,4]
 => [[1,2,3,4,13],[5,6,7,8],[9,10,11,12]]
 => ? = 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => [5,5,4]
 => [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
 => ? = 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [2,2,2,2,2,2,2]
 => [7,7]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
 => ? = 2
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => [3,2,2,2,2]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? = 2
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,2,2,2,2]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? = 2
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,2,2,2,2]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? = 2
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,2,2,2]
 => [5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ? = 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3]
 => [4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
 => ? = 3
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [3,3,2,2,2]
 => [5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => [6,6,2]
 => [[1,2,5,6,7,8],[3,4,11,12,13,14],[9,10]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => [5,5,3]
 => [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => [5,5,5]
 => [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
 => ? = 3
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => [5,5,4]
 => [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => [5,5,3]
 => [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
 => ? = 2
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3]
 => [4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
 => ? = 3
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2,2]
 => [6,6,3]
 => [[1,2,3,7,8,9],[4,5,6,13,14,15],[10,11,12]]
 => ? = 2
([(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]
 => [3,3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
 => ? = 4
([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3]
 => [4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
 => ? = 3
([(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]
 => [4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
 => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
 => [3,2,2,2,2,2]
 => [6,6,1]
 => [[1,3,4,5,6,7],[2,9,10,11,12,13],[8]]
 => ? = 2
([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2]
 => [5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => [5,5,3]
 => [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
 => ? = 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => [5,5,5]
 => [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
 => ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => [3,3,2,2,2]
 => [5,5,2]
 => [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
 => ? = 2
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => [5,5,4]
 => [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => [5,5,5]
 => [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
 => ? = 3
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,3,3,3]
 => [4,4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
 => ? = 3
([(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]
 => [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),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => [4,4,4,2]
 => [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
 => ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,4,4,4]
 => [4,4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
 => ? = 4
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => [2,2,2,2,2,1]
 => [6,5]
 => [[1,2,3,4,5,11],[6,7,8,9,10]]
 => ? = 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => [2,2,2,2,2,1]
 => [6,5]
 => [[1,2,3,4,5,11],[6,7,8,9,10]]
 => ? = 1
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [3,2,2,2,2]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? = 2
([(3,5),(3,6),(4,5),(4,6)],7)
 => [2,2,2,2,1,1,1]
 => [7,4]
 => [[1,2,3,4,9,10,11],[5,6,7,8]]
 => ? = 1

Description

The row containing the largest entry of a standard tableau.

Matching statistic: St000745

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 1
([(0,2),(1,2)],3)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
([(1,3),(2,3)],4)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 2
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 2
([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 3
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
([(2,4),(3,4)],5)
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 1
([(1,4),(2,4),(3,4)],5)
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 2
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 1
([(1,4),(2,3),(3,4)],5)
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => 1
([(0,1),(2,4),(3,4)],5)
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 2
([(1,3),(1,4),(2,3),(2,4)],5)
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => [[1,2,4,6,8],[3,5,7,9]]
 => 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]]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [[1,2,5],[3,6],[4,7]]
 => 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => [[1,2,5],[3,4,8],[6,7]]
 => [[1,3,6],[2,4,7],[5,8]]
 => 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]]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
 => 2
([(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]]
 => [[1,4,7],[2,5,8],[3,6,9]]
 => 3
([(0,4),(1,3),(2,3),(2,4)],5)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => [[1,3,5,7],[2,4,6,8]]
 => 2
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 3
([(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]]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 2
([(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]]
 => [[1,3,5,7],[2,4,6,8],[9]]
 => 2
([(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]]
 => [[1,4,7],[2,5,8],[3,6,9]]
 => 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]]
 => [[1,3,6],[2,4,7],[5,8]]
 => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 2
([(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]]
 => [[1,4],[2,5],[3,6],[7]]
 => 3
([(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]]
 => [[1,3,5,8],[2,4,6,9],[7,10]]
 => 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]]
 => [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
 => ? = 3
([(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]]
 => [[1,5],[2,6],[3,7],[4,8]]
 => 4
([],6)
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 1
([(3,5),(4,5)],6)
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7]]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [[1,2,3,5,7],[4,6,8]]
 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => [[1,2,4,6,8],[3,5,7,9]]
 => 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]]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 2
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 1
([(2,5),(3,4),(4,5)],6)
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [[1,2,3,5,7],[4,6,8]]
 => 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]]
 => [[1,2,4,6,8,10],[3,5,7,9,11]]
 => ? = 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
 => [[1,2,4,6,8,10,12],[3,5,7,9,11,13]]
 => ? = 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
 => [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
 => ? = 2
([(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]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => ? = 2
([(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]]
 => [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
 => ? = 3
([(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]]
 => [[1,2,4,6,8,10],[3,5,7,9,11]]
 => ? = 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]]
 => [[1,3,5,7,9],[2,4,6,8,10],[11]]
 => ? = 2
([(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]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => ? = 2
([(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]]
 => [[1,3,5,7,9],[2,4,6,8,10],[11]]
 => ? = 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
 => [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
 => ? = 2
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
 => [3,2,2,2,2,2]
 => [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12],[13]]
 => ? = 2
([(0,3),(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]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => ? = 2
([(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]]
 => [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
 => ? = 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]]
 => [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
 => ? = 3
([(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]]
 => [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => [[1,2,11],[3,4,14],[5,6],[7,8],[9,10],[12,13]]
 => [[1,3,5,7,9,12],[2,4,6,8,10,13],[11,14]]
 => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => [2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
 => [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
 => ? = 2
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
 => [[1,2,4,6,9],[3,5,7,10],[8,11]]
 => ? = 1
([(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]]
 => [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
 => ? = 2
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
 => [[1,2,5,8,11],[3,6,9,12],[4,7,10,13]]
 => ? = 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
 => [[1,3,6,9,12],[2,4,7,10,13],[5,8,11,14]]
 => ? = 2
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => [2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
 => [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]]
 => ? = 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]]
 => [[1,3,5,7,9],[2,4,6,8,10],[11]]
 => ? = 2
([(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]]
 => [[1,3,5,7,9],[2,4,6,8,10],[11]]
 => ? = 2
([(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]]
 => [[1,3,5,7,9],[2,4,6,8,10],[11]]
 => ? = 2
([(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]]
 => [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
 => ? = 2
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => ? = 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]]
 => [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
 => ? = 3
([(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]]
 => [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => [[1,2,11],[3,4,14],[5,6],[7,8],[9,10],[12,13]]
 => [[1,3,5,7,9,12],[2,4,6,8,10,13],[11,14]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
 => [[1,3,5,8,11],[2,4,6,9,12],[7,10,13]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
 => [[1,4,7,10,13],[2,5,8,11,14],[3,6,9,12,15]]
 => ? = 3
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
 => [[1,3,6,9,12],[2,4,7,10,13],[5,8,11,14]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
 => [[1,3,5,8,11],[2,4,6,9,12],[7,10,13]]
 => ? = 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]]
 => [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
 => ? = 3
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,4,3]
 => [[1,2,3,7],[4,5,6,11],[8,9,10]]
 => [[1,4,8],[2,5,9],[3,6,10],[7,11]]
 => ? = 3
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2,2]
 => [[1,2,9],[3,4,12],[5,6,15],[7,8],[10,11],[13,14]]
 => [[1,3,5,7,10,13],[2,4,6,8,11,14],[9,12,15]]
 => ? = 2
([(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]]
 => [[1,5,9],[2,6,10],[3,7,11],[4,8,12]]
 => ? = 4
([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10]]
 => [[1,3,6,9],[2,4,7,10],[5,8,11]]
 => ? = 2
([(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]]
 => [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
 => ? = 3
([(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]]
 => [[1,4,7,10],[2,5,8,11],[3,6,9,12]]
 => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6)
 => [3,2,2,2,2,2]
 => [[1,2,13],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => [[1,3,5,7,9,11],[2,4,6,8,10,12],[13]]
 => ? = 2
([(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]]
 => [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
 => ? = 2
([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
 => [[1,3,5,8,11],[2,4,6,9,12],[7,10,13]]
 => ? = 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
 => [[1,4,7,10,13],[2,5,8,11,14],[3,6,9,12,15]]
 => ? = 3
([(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]]
 => [[1,3,5,7,10],[2,4,6,8,11],[9,12]]
 => ? = 2
([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
 => [[1,3,6,9,12],[2,4,7,10,13],[5,8,11,14]]
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
 => [[1,4,7,10,13],[2,5,8,11,14],[3,6,9,12,15]]
 => ? = 3

Description

The index of the last row whose first entry is the row number in a standard Young tableau.

Matching statistic: St001184

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 83%

Values

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

Description

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

Matching statistic: St001185

Mapping

Mp00251: Graphs —clique sizes⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 83%

Values

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

Description

The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.

The following 13 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001481The minimal height of a peak of a Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001316The domatic number of a graph. St001621The number of atoms of a lattice. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition.