searching the database
Your data matches 9 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000160
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 1
[1,1]
=> 2
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 1
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 1
[4,1]
=> 1
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 2
[3,3]
=> 2
[3,2,1]
=> 1
[3,1,1,1]
=> 3
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
[7]
=> 1
[6,1]
=> 1
[5,2]
=> 1
[5,1,1]
=> 2
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 3
[3,3,1]
=> 1
[3,2,2]
=> 2
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 4
[2,2,2,1]
=> 1
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> 7
[8]
=> 1
[7,1]
=> 1
[6,2]
=> 1
[6,1,1]
=> 2
[5,3]
=> 1
[5,2,1]
=> 1
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part spt(λ) of a partition λ.
The sum spt(n)=∑λ⊢nspt(λ) satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000382
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 77% ●values known / values provided: 95%●distinct values known / distinct values provided: 77%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 77% ●values known / values provided: 95%●distinct values known / distinct values provided: 77%
Values
[1]
=> [1,0,1,0]
=> [1,1] => [1,1] => 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => [1,2] => 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,3] => 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,1,1] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [3,1] => 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,4] => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,2] => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [2,2] => 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,1,1] => 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [4,1] => 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [1,5] => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,3] => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,2] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,1,1] => 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,2,1] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [3,1,1] => 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [5,1] => 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [1,6] => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [1,1,4] => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,3] => 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [2,1,2] => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [2,3] => 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,1,1,1] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [3,1,1] => 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [3,2] => 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [2,2,1] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [4,1,1] => 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => [6,1] => 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,7] => 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => [1,1,5] => 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => [1,1,4] => 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => [2,1,3] => 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,3] => 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,1,2] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [3,1,1] => 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,2,2] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [2,1,2] => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [2,1,1,1] => 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [4,1,1] => 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,3,1] => 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [3,2,1] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => [5,1,1] => 5
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [7,1] => 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,8] => 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,6] => 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => [1,1,5] => 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => [2,1,4] => 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => [1,1,4] => 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => [1,1,1,3] => 1
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1,1] => ? => ? = 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [7,2,1] => [2,1,7] => ? = 2
[2,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1] => ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? => ? = 11
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? => ? => ? = 1
[11,1]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,1] => ? => ? = 1
[10,2]
=> [1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,1] => ? => ? = 1
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [8,2,1] => ? => ? = 2
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [6,3,1] => [3,1,6] => ? = 3
[3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0]
=> [1,9,1] => [9,1,1] => ? = 9
[2,2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0]
=> [1,8,2] => [8,2,1] => ? = 8
[2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [1,10,1] => [10,1,1] => ? = 10
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? => ? => ? = 12
[]
=> []
=> [] => ? => ? = 0
Description
The first part of an integer composition.
Matching statistic: St000326
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 69%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00280: Binary words —path rowmotion⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 64% ●values known / values provided: 64%●distinct values known / distinct values provided: 69%
Values
[1]
=> [1]
=> 10 => 11 => 1
[2]
=> [1,1]
=> 110 => 111 => 1
[1,1]
=> [2]
=> 100 => 011 => 2
[3]
=> [1,1,1]
=> 1110 => 1111 => 1
[2,1]
=> [2,1]
=> 1010 => 1101 => 1
[1,1,1]
=> [3]
=> 1000 => 0011 => 3
[4]
=> [1,1,1,1]
=> 11110 => 11111 => 1
[3,1]
=> [2,1,1]
=> 10110 => 11011 => 1
[2,2]
=> [2,2]
=> 1100 => 0111 => 2
[2,1,1]
=> [3,1]
=> 10010 => 01101 => 2
[1,1,1,1]
=> [4]
=> 10000 => 00011 => 4
[5]
=> [1,1,1,1,1]
=> 111110 => 111111 => 1
[4,1]
=> [2,1,1,1]
=> 101110 => 110111 => 1
[3,2]
=> [2,2,1]
=> 11010 => 11101 => 1
[3,1,1]
=> [3,1,1]
=> 100110 => 011011 => 2
[2,2,1]
=> [3,2]
=> 10100 => 11001 => 1
[2,1,1,1]
=> [4,1]
=> 100010 => 001101 => 3
[1,1,1,1,1]
=> [5]
=> 100000 => 000011 => 5
[6]
=> [1,1,1,1,1,1]
=> 1111110 => 1111111 => 1
[5,1]
=> [2,1,1,1,1]
=> 1011110 => 1101111 => 1
[4,2]
=> [2,2,1,1]
=> 110110 => 111011 => 1
[4,1,1]
=> [3,1,1,1]
=> 1001110 => 0110111 => 2
[3,3]
=> [2,2,2]
=> 11100 => 01111 => 2
[3,2,1]
=> [3,2,1]
=> 101010 => 110101 => 1
[3,1,1,1]
=> [4,1,1]
=> 1000110 => 0011011 => 3
[2,2,2]
=> [3,3]
=> 11000 => 00111 => 3
[2,2,1,1]
=> [4,2]
=> 100100 => 011001 => 2
[2,1,1,1,1]
=> [5,1]
=> 1000010 => 0001101 => 4
[1,1,1,1,1,1]
=> [6]
=> 1000000 => 0000011 => 6
[7]
=> [1,1,1,1,1,1,1]
=> 11111110 => 11111111 => 1
[6,1]
=> [2,1,1,1,1,1]
=> 10111110 => 11011111 => 1
[5,2]
=> [2,2,1,1,1]
=> 1101110 => 1110111 => 1
[5,1,1]
=> [3,1,1,1,1]
=> 10011110 => 01101111 => 2
[4,3]
=> [2,2,2,1]
=> 111010 => 111101 => 1
[4,2,1]
=> [3,2,1,1]
=> 1010110 => 1101011 => 1
[4,1,1,1]
=> [4,1,1,1]
=> 10001110 => 00110111 => 3
[3,3,1]
=> [3,2,2]
=> 101100 => 110011 => 1
[3,2,2]
=> [3,3,1]
=> 110010 => 011101 => 2
[3,2,1,1]
=> [4,2,1]
=> 1001010 => 0110101 => 2
[3,1,1,1,1]
=> [5,1,1]
=> 10000110 => 00011011 => 4
[2,2,2,1]
=> [4,3]
=> 101000 => 110001 => 1
[2,2,1,1,1]
=> [5,2]
=> 1000100 => 0011001 => 3
[2,1,1,1,1,1]
=> [6,1]
=> 10000010 => 00001101 => 5
[1,1,1,1,1,1,1]
=> [7]
=> 10000000 => 00000011 => 7
[8]
=> [1,1,1,1,1,1,1,1]
=> 111111110 => 111111111 => 1
[7,1]
=> [2,1,1,1,1,1,1]
=> 101111110 => 110111111 => 1
[6,2]
=> [2,2,1,1,1,1]
=> 11011110 => 11101111 => 1
[6,1,1]
=> [3,1,1,1,1,1]
=> 100111110 => 011011111 => 2
[5,3]
=> [2,2,2,1,1]
=> 1110110 => 1111011 => 1
[5,2,1]
=> [3,2,1,1,1]
=> 10101110 => 11010111 => 1
[9]
=> [1,1,1,1,1,1,1,1,1]
=> 1111111110 => 1111111111 => ? = 1
[8,1]
=> [2,1,1,1,1,1,1,1]
=> 1011111110 => 1101111111 => ? = 1
[6,1,1,1]
=> [4,1,1,1,1,1]
=> 1000111110 => 0011011111 => ? = 3
[10]
=> [1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => 11111111111 => ? = 1
[9,1]
=> [2,1,1,1,1,1,1,1,1]
=> 10111111110 => 11011111111 => ? = 1
[8,2]
=> [2,2,1,1,1,1,1,1]
=> 1101111110 => 1110111111 => ? = 1
[8,1,1]
=> [3,1,1,1,1,1,1,1]
=> 10011111110 => 01101111111 => ? = 2
[7,2,1]
=> [3,2,1,1,1,1,1]
=> 1010111110 => 1101011111 => ? = 1
[7,1,1,1]
=> [4,1,1,1,1,1,1]
=> 10001111110 => 00110111111 => ? = 3
[6,2,1,1]
=> [4,2,1,1,1,1]
=> 1001011110 => 0110101111 => ? = 2
[6,1,1,1,1]
=> [5,1,1,1,1,1]
=> 10000111110 => 00011011111 => ? = 4
[5,2,1,1,1]
=> [5,2,1,1,1]
=> 1000101110 => 0011010111 => ? = 3
[5,1,1,1,1,1]
=> [6,1,1,1,1]
=> 10000011110 => 00001101111 => ? = 5
[4,1,1,1,1,1,1]
=> [7,1,1,1]
=> 10000001110 => 00000110111 => ? = 6
[3,1,1,1,1,1,1,1]
=> [8,1,1]
=> 10000000110 => 00000011011 => ? = 7
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> 10000000000 => 00000000011 => ? = 10
[11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? = 1
[10,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? = 1
[9,2]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => 11101111111 => ? = 1
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? = 2
[8,3]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 1111011111 => ? = 1
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> 10101111110 => 11010111111 => ? = 1
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? = 3
[7,3,1]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? => ? = 1
[7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? => ? = 2
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> 10010111110 => 01101011111 => ? = 2
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> 100001111110 => ? => ? = 4
[6,3,1,1]
=> [4,2,2,1,1,1]
=> 1001101110 => ? => ? = 2
[6,2,2,1]
=> [4,3,1,1,1,1]
=> 1010011110 => ? => ? = 1
[6,2,1,1,1]
=> [5,2,1,1,1,1]
=> 10001011110 => 00110101111 => ? = 3
[6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> 100000111110 => ? => ? = 5
[5,3,1,1,1]
=> [5,2,2,1,1]
=> 1000110110 => 0011011011 => ? = 3
[5,2,2,1,1]
=> [5,3,1,1,1]
=> 1001001110 => 0110010111 => ? = 2
[5,2,1,1,1,1]
=> [6,2,1,1,1]
=> 10000101110 => 00011010111 => ? = 4
[5,1,1,1,1,1,1]
=> [7,1,1,1,1]
=> 100000011110 => ? => ? = 6
[4,3,1,1,1,1]
=> [6,2,2,1]
=> 1000011010 => ? => ? = 4
[4,2,2,1,1,1]
=> [6,3,1,1]
=> 1000100110 => ? => ? = 3
[4,2,1,1,1,1,1]
=> [7,2,1,1]
=> 10000010110 => 00001101011 => ? = 5
[4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> 100000001110 => ? => ? = 7
[3,3,1,1,1,1,1]
=> [7,2,2]
=> 1000001100 => ? => ? = 5
[3,2,2,1,1,1,1]
=> [7,3,1]
=> 1000010010 => ? => ? = 4
[3,2,1,1,1,1,1,1]
=> [8,2,1]
=> 10000001010 => 00000110101 => ? = 6
[3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> 100000000110 => ? => ? = 8
[2,2,2,1,1,1,1,1]
=> [8,3]
=> 1000001000 => ? => ? = 5
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> 10000000100 => 00000011001 => ? = 7
[2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> 100000000010 => ? => ? = 9
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> 100000000000 => ? => ? = 11
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? => ? = 1
[11,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? => ? = 1
[10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? => ? = 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: St000383
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 62%
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 62%
Values
[1]
=> [[1]]
=> [1] => [1] => 1
[2]
=> [[1,2]]
=> [2] => [1] => 1
[1,1]
=> [[1],[2]]
=> [1,1] => [2] => 2
[3]
=> [[1,2,3]]
=> [3] => [1] => 1
[2,1]
=> [[1,2],[3]]
=> [2,1] => [1,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => [3] => 3
[4]
=> [[1,2,3,4]]
=> [4] => [1] => 1
[3,1]
=> [[1,2,3],[4]]
=> [3,1] => [1,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => [2] => 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => [1,2] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => [4] => 4
[5]
=> [[1,2,3,4,5]]
=> [5] => [1] => 1
[4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => [1,1] => 1
[3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => [1,1] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => [1,2] => 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => [2,1] => 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => [1,3] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => [5] => 5
[6]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => [1,1] => 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => [1,1] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => [1,2] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [2] => 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => [1,1,1] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => [1,3] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => [3] => 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => [2,2] => 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => [1,4] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => [6] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => [1] => 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [6,1] => [1,1] => 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [5,2] => [1,1] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [5,1,1] => [1,2] => 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [4,3] => [1,1] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [4,2,1] => [1,1,1] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [4,1,1,1] => [1,3] => 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [3,3,1] => [2,1] => 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [3,2,2] => [1,2] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [3,2,1,1] => [1,1,2] => 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [3,1,1,1,1] => [1,4] => 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [2,2,2,1] => [3,1] => 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [2,2,1,1,1] => [2,3] => 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [2,1,1,1,1,1] => [1,5] => 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => [7] => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => [1] => 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [7,1] => [1,1] => 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [6,2] => [1,1] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [6,1,1] => [1,2] => 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [5,3] => [1,1] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [5,2,1] => [1,1,1] => 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [1,1,1,1,1,1,1,1] => [8] => ? = 8
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [1,1,1,1,1,1,1,1,1] => [9] => ? = 9
[2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [2,2,1,1,1,1,1,1] => [2,6] => ? = 6
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [1,1,1,1,1,1,1,1,1,1] => [10] => ? = 10
[11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? => ? = 1
[10,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ? => ? = 1
[9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? => ? => ? = 1
[9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? => ? => ? = 2
[8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? => ? => ? = 1
[8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? => ? => ? = 1
[8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? => ? = 3
[7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? => ? => ? = 1
[7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? => ? => ? = 1
[7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? => ? => ? = 2
[7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? => ? => ? = 2
[7,1,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10],[11]]
=> ? => ? => ? = 4
[6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? => ? => ? = 1
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ? => ? = 1
[6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? => ? => ? = 1
[6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? => ? => ? = 2
[6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? => ? => ? = 1
[6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? = 3
[6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 5
[5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? => ? => ? = 1
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [5,4,2] => ? => ? = 1
[5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [5,4,1,1] => ? => ? = 2
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [5,3,3] => ? => ? = 2
[5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [5,3,2,1] => ? => ? = 1
[5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [5,3,1,1,1] => ? => ? = 3
[5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [5,2,2,2] => ? => ? = 3
[5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [5,2,2,1,1] => ? => ? = 2
[5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? => ? = 4
[5,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 6
[4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [4,4,2,1] => ? => ? = 1
[4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [4,4,1,1,1] => ? => ? = 3
[4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [4,3,3,1] => ? => ? = 1
[4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [4,3,2,2] => ? => ? = 2
[4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [4,3,2,1,1] => ? => ? = 2
[4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? => ? => ? = 4
[4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [4,2,2,2,1] => ? => ? = 1
[4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? = 3
[4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 5
[4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 7
[3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [3,3,3,1,1] => ? => ? = 2
[3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [3,3,2,2,1] => ? => ? = 1
[3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? => ? = 3
[3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 5
[3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? => ? = 4
[3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? => ? = 2
[3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? => ? = 4
Description
The last part of an integer composition.
Matching statistic: St000993
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 62%
Mp00189: Skew partitions —rotate⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 48% ●values known / values provided: 48%●distinct values known / distinct values provided: 62%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> []
=> ? = 1
[2]
=> [[2],[]]
=> [[2],[]]
=> []
=> ? = 1
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> []
=> ? = 2
[3]
=> [[3],[]]
=> [[3],[]]
=> []
=> ? = 1
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [1]
=> ? = 1
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> []
=> ? = 3
[4]
=> [[4],[]]
=> [[4],[]]
=> []
=> ? = 1
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [2]
=> 1
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> []
=> ? = 2
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 2
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> []
=> ? = 4
[5]
=> [[5],[]]
=> [[5],[]]
=> []
=> ? = 1
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [3]
=> 1
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [1]
=> ? = 1
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [2,2]
=> 2
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [1]
=> ? = 1
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> []
=> ? = 5
[6]
=> [[6],[]]
=> [[6],[]]
=> []
=> ? = 1
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [4]
=> 1
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [2]
=> 1
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [3,3]
=> 2
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> []
=> ? = 2
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [2,1]
=> 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 3
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> []
=> ? = 3
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? = 6
[7]
=> [[7],[]]
=> [[7],[]]
=> []
=> ? = 1
[6,1]
=> [[6,1],[]]
=> [[6,6],[5]]
=> [5]
=> 1
[5,2]
=> [[5,2],[]]
=> [[5,5],[3]]
=> [3]
=> 1
[5,1,1]
=> [[5,1,1],[]]
=> [[5,5,5],[4,4]]
=> [4,4]
=> 2
[4,3]
=> [[4,3],[]]
=> [[4,4],[1]]
=> [1]
=> ? = 1
[4,2,1]
=> [[4,2,1],[]]
=> [[4,4,4],[3,2]]
=> [3,2]
=> 1
[4,1,1,1]
=> [[4,1,1,1],[]]
=> [[4,4,4,4],[3,3,3]]
=> [3,3,3]
=> 3
[3,3,1]
=> [[3,3,1],[]]
=> [[3,3,3],[2]]
=> [2]
=> 1
[3,2,2]
=> [[3,2,2],[]]
=> [[3,3,3],[1,1]]
=> [1,1]
=> 2
[3,2,1,1]
=> [[3,2,1,1],[]]
=> [[3,3,3,3],[2,2,1]]
=> [2,2,1]
=> 2
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,2]]
=> [2,2,2,2]
=> 4
[2,2,2,1]
=> [[2,2,2,1],[]]
=> [[2,2,2,2],[1]]
=> [1]
=> ? = 1
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 3
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1,1]]
=> [1,1,1,1,1]
=> 5
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1],[]]
=> []
=> ? = 7
[8]
=> [[8],[]]
=> [[8],[]]
=> []
=> ? = 1
[7,1]
=> [[7,1],[]]
=> [[7,7],[6]]
=> [6]
=> 1
[6,2]
=> [[6,2],[]]
=> [[6,6],[4]]
=> [4]
=> 1
[6,1,1]
=> [[6,1,1],[]]
=> [[6,6,6],[5,5]]
=> [5,5]
=> 2
[5,3]
=> [[5,3],[]]
=> [[5,5],[2]]
=> [2]
=> 1
[5,2,1]
=> [[5,2,1],[]]
=> [[5,5,5],[4,3]]
=> [4,3]
=> 1
[5,1,1,1]
=> [[5,1,1,1],[]]
=> [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> 3
[4,4]
=> [[4,4],[]]
=> [[4,4],[]]
=> []
=> ? = 2
[4,3,1]
=> [[4,3,1],[]]
=> [[4,4,4],[3,1]]
=> [3,1]
=> 1
[4,2,2]
=> [[4,2,2],[]]
=> [[4,4,4],[2,2]]
=> [2,2]
=> 2
[4,2,1,1]
=> [[4,2,1,1],[]]
=> [[4,4,4,4],[3,3,2]]
=> [3,3,2]
=> 2
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> 4
[3,3,2]
=> [[3,3,2],[]]
=> [[3,3,3],[1]]
=> [1]
=> ? = 1
[3,3,1,1]
=> [[3,3,1,1],[]]
=> [[3,3,3,3],[2,2]]
=> [2,2]
=> 2
[3,2,2,1]
=> [[3,2,2,1],[]]
=> [[3,3,3,3],[2,1,1]]
=> [2,1,1]
=> 1
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> [[3,3,3,3,3],[2,2,2,1]]
=> [2,2,2,1]
=> 3
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [2,2,2,2,2]
=> 5
[2,2,2,2]
=> [[2,2,2,2],[]]
=> [[2,2,2,2],[]]
=> []
=> ? = 4
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> [[2,2,2,2,2],[1,1]]
=> [1,1]
=> 2
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> [[2,2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 4
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [1,1,1,1,1,1]
=> 6
[1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1,1],[]]
=> []
=> ? = 8
[9]
=> [[9],[]]
=> [[9],[]]
=> []
=> ? = 1
[8,1]
=> [[8,1],[]]
=> [[8,8],[7]]
=> [7]
=> 1
[7,2]
=> [[7,2],[]]
=> [[7,7],[5]]
=> [5]
=> 1
[7,1,1]
=> [[7,1,1],[]]
=> [[7,7,7],[6,6]]
=> [6,6]
=> 2
[6,3]
=> [[6,3],[]]
=> [[6,6],[3]]
=> [3]
=> 1
[6,2,1]
=> [[6,2,1],[]]
=> [[6,6,6],[5,4]]
=> [5,4]
=> 1
[6,1,1,1]
=> [[6,1,1,1],[]]
=> [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> ? = 3
[5,4]
=> [[5,4],[]]
=> [[5,5],[1]]
=> [1]
=> ? = 1
[5,3,1]
=> [[5,3,1],[]]
=> [[5,5,5],[4,2]]
=> [4,2]
=> 1
[5,2,2]
=> [[5,2,2],[]]
=> [[5,5,5],[3,3]]
=> [3,3]
=> 2
[5,2,1,1]
=> [[5,2,1,1],[]]
=> [[5,5,5,5],[4,4,3]]
=> [4,4,3]
=> 2
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> ? = 4
[4,4,1]
=> [[4,4,1],[]]
=> [[4,4,4],[3]]
=> [3]
=> 1
[4,3,2]
=> [[4,3,2],[]]
=> [[4,4,4],[2,1]]
=> [2,1]
=> 1
[4,1,1,1,1,1]
=> [[4,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> ? = 5
[3,3,3]
=> [[3,3,3],[]]
=> [[3,3,3],[]]
=> []
=> ? = 3
[2,2,2,2,1]
=> [[2,2,2,2,1],[]]
=> [[2,2,2,2,2],[1]]
=> [1]
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1,1,1],[]]
=> []
=> ? = 9
[10]
=> [[10],[]]
=> [[10],[]]
=> []
=> ? = 1
[8,1,1]
=> [[8,1,1],[]]
=> [[8,8,8],[7,7]]
=> [7,7]
=> ? = 2
[7,1,1,1]
=> [[7,1,1,1],[]]
=> [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> ? = 3
[6,2,1,1]
=> [[6,2,1,1],[]]
=> [[6,6,6,6],[5,5,4]]
=> [5,5,4]
=> ? = 2
[6,1,1,1,1]
=> [[6,1,1,1,1],[]]
=> [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> ? = 4
[5,5]
=> [[5,5],[]]
=> [[5,5],[]]
=> []
=> ? = 2
[5,2,1,1,1]
=> [[5,2,1,1,1],[]]
=> [[5,5,5,5,5],[4,4,4,3]]
=> [4,4,4,3]
=> ? = 3
[5,1,1,1,1,1]
=> [[5,1,1,1,1,1],[]]
=> [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> ? = 5
[4,2,1,1,1,1]
=> [[4,2,1,1,1,1],[]]
=> [[4,4,4,4,4,4],[3,3,3,3,2]]
=> [3,3,3,3,2]
=> ? = 4
[4,1,1,1,1,1,1]
=> [[4,1,1,1,1,1,1],[]]
=> [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> ? = 6
[3,1,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1,1],[]]
=> [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> ? = 7
[2,2,2,2,2]
=> [[2,2,2,2,2],[]]
=> [[2,2,2,2,2],[]]
=> []
=> ? = 5
[1,1,1,1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1,1,1,1,1],[]]
=> []
=> ? = 10
[11]
=> [[11],[]]
=> [[11],[]]
=> ?
=> ? = 1
[10,1]
=> [[10,1],[]]
=> [[10,10],[9]]
=> ?
=> ? = 1
[9,2]
=> [[9,2],[]]
=> ?
=> ?
=> ? = 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000907
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 54%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000907: Posets ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 54%
Values
[1]
=> [1,0]
=> [1,0]
=> ([],1)
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> ([],2)
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 2
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 3
[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]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ([],6)
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 2
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(4,1),(4,2)],5)
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 3
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 2
[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]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 4
[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]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ([],7)
=> 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6)],7)
=> 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5)],6)
=> 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ([(0,6),(6,1),(6,2),(6,3),(6,4),(6,5)],7)
=> 2
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ([(0,5),(5,6),(6,1),(6,2),(6,3),(6,4)],7)
=> 3
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ([(0,5),(4,2),(4,3),(5,1),(5,4)],6)
=> 2
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> 4
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[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]
=> ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 3
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> 5
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 7
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ([],8)
=> 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7)],8)
=> ? = 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ([(0,7),(7,1),(7,2),(7,3),(7,4),(7,5),(7,6)],8)
=> ? = 2
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> ([(2,3),(2,4),(2,5)],6)
=> 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> ([(0,5),(0,6),(6,1),(6,2),(6,3),(6,4)],7)
=> 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ([(0,6),(6,7),(7,1),(7,2),(7,3),(7,4),(7,5)],8)
=> ? = 3
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> 2
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(5,7),(6,5),(7,1),(7,2),(7,3),(7,4)],8)
=> ? = 4
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(4,5),(5,7),(6,4),(7,1),(7,2),(7,3)],8)
=> ? = 5
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(3,5),(4,3),(5,7),(6,4),(7,1),(7,2)],8)
=> ? = 6
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 8
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ([],9)
=> ? = 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8)],9)
=> ? = 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ? = 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ([(0,8),(8,1),(8,2),(8,3),(8,4),(8,5),(8,6),(8,7)],9)
=> ? = 2
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> ([(0,6),(0,7),(7,1),(7,2),(7,3),(7,4),(7,5)],8)
=> ? = 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ([(0,7),(7,8),(8,1),(8,2),(8,3),(8,4),(8,5),(8,6)],9)
=> ? = 3
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 2
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> ([(0,6),(6,5),(6,7),(7,1),(7,2),(7,3),(7,4)],8)
=> ? = 2
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(6,8),(7,6),(8,1),(8,2),(8,3),(8,4),(8,5)],9)
=> ? = 4
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> ([(0,5),(5,6),(6,4),(6,7),(7,1),(7,2),(7,3)],8)
=> ? = 3
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(5,6),(6,8),(7,5),(8,1),(8,2),(8,3),(8,4)],9)
=> ? = 5
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(2,6),(3,1),(3,5),(3,6),(4,2),(4,3)],7)
=> ? = 2
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(4,7),(5,4),(6,2),(6,3),(7,1),(7,6)],8)
=> ? = 4
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(4,6),(5,4),(6,8),(7,5),(8,1),(8,2),(8,3)],9)
=> ? = 6
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
=> ? = 3
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
=> ? = 5
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,7),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(8,2)],9)
=> ? = 7
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 9
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ([],10)
=> ? = 1
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9)],10)
=> ? = 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8)],9)
=> ? = 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ([(0,9),(9,1),(9,2),(9,3),(9,4),(9,5),(9,6),(9,7),(9,8)],10)
=> ? = 2
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> ([(2,3),(2,4),(2,5),(2,6),(2,7)],8)
=> ? = 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> ([(0,7),(0,8),(8,1),(8,2),(8,3),(8,4),(8,5),(8,6)],9)
=> ? = 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ([(0,8),(8,9),(9,1),(9,2),(9,3),(9,4),(9,5),(9,6),(9,7)],10)
=> ? = 3
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ([(0,5),(0,6),(0,7),(7,1),(7,2),(7,3),(7,4)],8)
=> ? = 1
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7)],8)
=> ? = 2
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> ([(0,7),(7,6),(7,8),(8,1),(8,2),(8,3),(8,4),(8,5)],9)
=> ? = 2
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,8),(7,9),(8,7),(9,1),(9,2),(9,3),(9,4),(9,5),(9,6)],10)
=> ? = 4
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6)],7)
=> ? = 1
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ([(0,7),(6,3),(6,4),(6,5),(7,1),(7,2),(7,6)],8)
=> ? = 2
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(2,5),(2,6),(2,7),(3,1),(3,4),(3,5),(3,6),(3,7)],8)
=> ? = 1
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> ([(0,6),(6,7),(7,5),(7,8),(8,1),(8,2),(8,3),(8,4)],9)
=> ? = 3
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,8),(6,7),(7,9),(8,6),(9,1),(9,2),(9,3),(9,4),(9,5)],10)
=> ? = 5
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> ([(0,5),(5,7),(6,3),(6,4),(7,1),(7,2),(7,6)],8)
=> ? = 3
[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]
=> ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5)],7)
=> ? = 3
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(2,5),(2,6),(2,7),(3,1),(3,5),(3,6),(3,7),(4,2),(4,3)],8)
=> ? = 2
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(5,7),(6,5),(7,4),(7,8),(8,1),(8,2),(8,3)],9)
=> ? = 4
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,8),(5,7),(6,9),(7,6),(8,5),(9,1),(9,2),(9,3),(9,4)],10)
=> ? = 6
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0]
=> ([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
=> ? = 2
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ([(0,6),(4,7),(5,3),(6,4),(7,1),(7,2),(7,5)],8)
=> ? = 4
[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]
=> ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,1),(3,6)],7)
=> ? = 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ([(0,4),(2,6),(2,7),(3,1),(3,6),(3,7),(4,5),(5,2),(5,3)],8)
=> ? = 3
Description
The number of maximal antichains of minimal length in a poset.
Matching statistic: St000153
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 77%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000153: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 77%
Values
[1]
=> [[1]]
=> {{1}}
=> [1] => 1
[2]
=> [[1,2]]
=> {{1,2}}
=> [2,1] => 1
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> [1,2] => 2
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> [2,3,1] => 1
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> [3,2,1] => 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> [1,2,3] => 3
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> [2,3,4,1] => 1
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> [3,2,4,1] => 1
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> [2,1,4,3] => 2
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> [4,2,3,1] => 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => 4
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> [2,3,4,5,1] => 1
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => 1
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> [2,5,4,3,1] => 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => 5
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> [4,2,3,5,6,1] => 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> [2,3,1,5,6,4] => 2
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> [3,5,6,4,2,1] => 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> [5,2,3,4,6,1] => 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => 3
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> [4,6,3,1,5,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> [6,2,3,4,5,1] => 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> [1,2,3,4,5,6] => 6
[7]
=> [[1,2,3,4,5,6,7]]
=> {{1,2,3,4,5,6,7}}
=> [2,3,4,5,6,7,1] => 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> {{1,3,4,5,6,7},{2}}
=> [3,2,4,5,6,7,1] => 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> {{1,2,5,6,7},{3,4}}
=> [2,5,4,3,6,7,1] => ? = 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> {{1,4,5,6,7},{2},{3}}
=> [4,2,3,5,6,7,1] => 2
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> {{1,2,3,7},{4,5,6}}
=> [2,3,7,5,6,4,1] => ? = 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> {{1,3,6,7},{2,5},{4}}
=> [3,5,6,4,2,7,1] => ? = 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> {{1,5,6,7},{2},{3},{4}}
=> [5,2,3,4,6,7,1] => 3
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> {{1,3,4},{2,6,7},{5}}
=> [3,6,4,1,5,7,2] => ? = 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> {{1,2,7},{3,4},{5,6}}
=> [2,7,4,3,6,5,1] => ? = 2
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> {{1,4,7},{2,6},{3},{5}}
=> [4,6,3,7,5,2,1] => ? = 2
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> {{1,6,7},{2},{3},{4},{5}}
=> [6,2,3,4,5,7,1] => 4
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> {{1,3},{2,5},{4,7},{6}}
=> [3,5,1,7,2,6,4] => ? = 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> {{1,5},{2,7},{3},{4},{6}}
=> [5,7,3,4,1,6,2] => ? = 3
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> {{1,7},{2},{3},{4},{5},{6}}
=> [7,2,3,4,5,6,1] => ? = 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> [1,2,3,4,5,6,7] => 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> {{1,2,3,4,5,6,7,8}}
=> [2,3,4,5,6,7,8,1] => 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> {{1,3,4,5,6,7,8},{2}}
=> [3,2,4,5,6,7,8,1] => 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> {{1,2,5,6,7,8},{3,4}}
=> [2,5,4,3,6,7,8,1] => ? = 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> {{1,4,5,6,7,8},{2},{3}}
=> [4,2,3,5,6,7,8,1] => 2
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> {{1,2,3,7,8},{4,5,6}}
=> [2,3,7,5,6,4,8,1] => ? = 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> {{1,3,6,7,8},{2,5},{4}}
=> [3,5,6,4,2,7,8,1] => ? = 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> {{1,5,6,7,8},{2},{3},{4}}
=> [5,2,3,4,6,7,8,1] => 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> [2,3,4,1,6,7,8,5] => 2
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> [3,6,4,8,5,7,2,1] => ? = 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> [2,7,4,3,6,5,8,1] => ? = 2
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> [4,6,3,7,5,2,8,1] => ? = 2
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> [6,2,3,4,5,7,8,1] => ? = 4
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> [2,5,4,8,1,7,6,3] => ? = 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> {{1,4,5},{2,7,8},{3},{6}}
=> [4,7,3,5,1,6,8,2] => ? = 2
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> [3,5,8,7,2,6,4,1] => ? = 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> {{1,5,8},{2,7},{3},{4},{6}}
=> [5,7,3,4,8,6,2,1] => ? = 3
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> [7,2,3,4,5,6,8,1] => ? = 5
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> [2,1,4,3,6,5,8,7] => 4
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> [4,6,8,1,5,2,7,3] => ? = 2
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> [6,8,3,4,5,1,7,2] => ? = 4
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> [8,2,3,4,5,6,7,1] => ? = 6
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,2,3,4,5,6,7,8] => 8
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> {{1,2,3,4,5,6,7,8,9}}
=> [2,3,4,5,6,7,8,9,1] => 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> {{1,3,4,5,6,7,8,9},{2}}
=> [3,2,4,5,6,7,8,9,1] => 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> {{1,2,5,6,7,8,9},{3,4}}
=> [2,5,4,3,6,7,8,9,1] => ? = 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> {{1,4,5,6,7,8,9},{2},{3}}
=> [4,2,3,5,6,7,8,9,1] => 2
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> {{1,2,3,7,8,9},{4,5,6}}
=> [2,3,7,5,6,4,8,9,1] => ? = 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> {{1,3,6,7,8,9},{2,5},{4}}
=> [3,5,6,4,2,7,8,9,1] => ? = 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> {{1,5,6,7,8,9},{2},{3},{4}}
=> [5,2,3,4,6,7,8,9,1] => ? = 3
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> {{1,2,3,4,9},{5,6,7,8}}
=> [2,3,4,9,6,7,8,5,1] => ? = 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> {{1,3,4,8,9},{2,6,7},{5}}
=> [3,6,4,8,5,7,2,9,1] => ? = 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> {{1,2,7,8,9},{3,4},{5,6}}
=> [2,7,4,3,6,5,8,9,1] => ? = 2
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> {{1,4,7,8,9},{2,6},{3},{5}}
=> [4,6,3,7,5,2,8,9,1] => ? = 2
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> {{1,6,7,8,9},{2},{3},{4},{5}}
=> [6,2,3,4,5,7,8,9,1] => ? = 4
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> {{1,3,4,5},{2,7,8,9},{6}}
=> [3,7,4,5,1,6,8,9,2] => ? = 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> {{1,2,5,9},{3,4,8},{6,7}}
=> [2,5,4,8,9,7,6,3,1] => ? = 1
[4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> {{1,4,5,9},{2,7,8},{3},{6}}
=> [4,7,3,5,9,6,8,2,1] => ? = 2
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> {{1,3,8,9},{2,5},{4,7},{6}}
=> [3,5,8,7,2,6,4,9,1] => ? = 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> {{1,5,8,9},{2,7},{3},{4},{6}}
=> [5,7,3,4,8,6,2,9,1] => ? = 3
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> {{1,7,8,9},{2},{3},{4},{5},{6}}
=> [7,2,3,4,5,6,8,9,1] => ? = 5
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> [2,3,1,5,6,4,8,9,7] => ? = 3
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> {{1,3,6},{2,5,9},{4,8},{7}}
=> [3,5,6,8,9,1,7,4,2] => ? = 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> {{1,5,6},{2,8,9},{3},{4},{7}}
=> [5,8,3,4,6,1,7,9,2] => ? = 3
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> {{1,2,9},{3,4},{5,6},{7,8}}
=> [2,9,4,3,6,5,8,7,1] => ? = 3
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> {{1,4,9},{2,6},{3,8},{5},{7}}
=> [4,6,8,9,5,2,7,3,1] => ? = 2
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> {{1,6,9},{2,8},{3},{4},{5},{7}}
=> [6,8,3,4,5,9,7,2,1] => ? = 4
[3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> [8,2,3,4,5,6,7,9,1] => ? = 6
[2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> {{1,3},{2,5},{4,7},{6,9},{8}}
=> [3,5,1,7,2,9,4,8,6] => ? = 1
[2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> {{1,5},{2,7},{3,9},{4},{6},{8}}
=> [5,7,9,4,1,6,2,8,3] => ? = 3
[2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> {{1,7},{2,9},{3},{4},{5},{6},{8}}
=> [7,9,3,4,5,6,1,8,2] => ? = 5
[2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> [9,2,3,4,5,6,7,8,1] => ? = 7
[1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [1,2,3,4,5,6,7,8,9] => 9
[10]
=> [[1,2,3,4,5,6,7,8,9,10]]
=> {{1,2,3,4,5,6,7,8,9,10}}
=> [2,3,4,5,6,7,8,9,10,1] => 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> {{1,3,4,5,6,7,8,9,10},{2}}
=> [3,2,4,5,6,7,8,9,10,1] => 1
[2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> {{1,2},{3,4},{5,6},{7,8},{9,10}}
=> [2,1,4,3,6,5,8,7,10,9] => 5
[1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,2,3,4,5,6,7,8,9,10] => 10
Description
The number of adjacent cycles of a permutation.
This is the number of cycles of the permutation of the form (i,i+1,i+2,...i+k) which includes the fixed points (i).
Matching statistic: St000910
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000910: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 54%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000910: Posets ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 54%
Values
[1]
=> [[1]]
=> [1] => ([],1)
=> ? = 1
[2]
=> [[1,2]]
=> [1,2] => ([(0,1)],2)
=> 1
[1,1]
=> [[1],[2]]
=> [2,1] => ([],2)
=> 2
[3]
=> [[1,2,3]]
=> [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => ([(1,2)],3)
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => ([],3)
=> 3
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(2,3)],4)
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => ([],4)
=> 4
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 2
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(3,4)],5)
=> 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([],5)
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
=> 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
=> 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
=> 2
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
=> 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
=> 3
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
=> 2
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => ([(4,5)],6)
=> 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([],6)
=> 6
[7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 2
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ([(1,6),(2,4),(5,3),(6,5)],7)
=> ? = 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
=> 3
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ([(1,6),(2,5),(5,3),(6,4)],7)
=> ? = 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ([(2,4),(3,5),(5,6)],7)
=> ? = 2
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ([(4,5),(5,6)],7)
=> 4
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ([(1,6),(2,5),(3,4)],7)
=> ? = 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ([(3,6),(4,5)],7)
=> 3
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ([(5,6)],7)
=> 5
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([],7)
=> 7
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
=> ? = 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => ([(1,7),(3,4),(4,6),(5,3),(6,2),(7,5)],8)
=> ? = 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => ([(0,7),(1,3),(4,6),(5,4),(6,2),(7,5)],8)
=> ? = 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => ([(2,7),(4,6),(5,4),(6,3),(7,5)],8)
=> ? = 2
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => ([(0,7),(1,6),(4,5),(5,3),(6,4),(7,2)],8)
=> ? = 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => ([(1,7),(2,4),(5,6),(6,3),(7,5)],8)
=> ? = 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => ([(3,4),(4,7),(6,5),(7,6)],8)
=> ? = 3
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => ([(0,7),(1,6),(4,2),(5,3),(6,4),(7,5)],8)
=> ? = 2
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => ([(1,6),(2,7),(5,4),(6,5),(7,3)],8)
=> ? = 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => ([(0,5),(1,4),(2,7),(6,3),(7,6)],8)
=> ? = 2
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => ([(2,4),(3,5),(5,6),(6,7)],8)
=> ? = 2
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => ([(4,5),(5,7),(7,6)],8)
=> ? = 4
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ([(0,5),(1,7),(2,6),(6,3),(7,4)],8)
=> ? = 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ([(2,5),(3,4),(4,6),(5,7)],8)
=> ? = 2
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ([(1,5),(2,4),(3,6),(6,7)],8)
=> ? = 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ([(3,5),(4,6),(6,7)],8)
=> ? = 3
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => ([(5,6),(6,7)],8)
=> ? = 5
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ([(0,7),(1,6),(2,5),(3,4)],8)
=> ? = 4
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ([(2,7),(3,6),(4,5)],8)
=> ? = 2
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ([(4,7),(5,6)],8)
=> ? = 4
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ([(6,7)],8)
=> ? = 6
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ([],8)
=> ? = 8
[9]
=> [[1,2,3,4,5,6,7,8,9]]
=> [1,2,3,4,5,6,7,8,9] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ? = 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => ([(1,8),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6)],9)
=> ? = 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => ([(0,8),(1,3),(4,5),(5,7),(6,4),(7,2),(8,6)],9)
=> ? = 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => ([(2,8),(4,5),(5,7),(6,4),(7,3),(8,6)],9)
=> ? = 2
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => ([(0,8),(1,7),(4,6),(5,4),(6,3),(7,5),(8,2)],9)
=> ? = 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => ([(1,8),(2,4),(5,7),(6,5),(7,3),(8,6)],9)
=> ? = 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => ([(3,8),(5,7),(6,5),(7,4),(8,6)],9)
=> ? = 3
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => ([(0,7),(1,8),(4,5),(5,2),(6,3),(7,6),(8,4)],9)
=> ? = 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => ([(1,8),(2,7),(5,6),(6,4),(7,5),(8,3)],9)
=> ? = 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => ([(0,5),(1,4),(2,8),(6,7),(7,3),(8,6)],9)
=> ? = 2
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => ([(2,8),(3,5),(6,7),(7,4),(8,6)],9)
=> ? = 2
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => ([(4,5),(5,8),(7,6),(8,7)],9)
=> ? = 4
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => ([(1,8),(2,7),(5,3),(6,4),(7,5),(8,6)],9)
=> ? = 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => ([(0,5),(1,7),(2,8),(6,3),(7,4),(8,6)],9)
=> ? = 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => ([(2,7),(3,8),(6,5),(7,6),(8,4)],9)
=> ? = 2
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => ([(1,6),(2,5),(3,8),(7,4),(8,7)],9)
=> ? = 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => ([(3,5),(4,6),(6,7),(7,8)],9)
=> ? = 3
Description
The number of maximal chains of minimal length in a poset.
Matching statistic: St001232
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 46%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 46%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 4
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 5
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 6
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ? = 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ? = 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 5
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 7
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> ? = 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 4
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ? = 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? = 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> ? = 3
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> ? = 5
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 4
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 4
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 8
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 3
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 2
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> 5
[4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> 3
[3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 4
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!