Your data matches 16 different statistics following compositions of up to 3 maps.
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Mp00202: Integer partitions first row removalInteger partitions
Mp00317: Integer partitions odd partsBinary words
Mp00105: Binary words complementBinary words
St001491: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[2,2]
=> [2]
=> 0 => 1 => 1
[3,2]
=> [2]
=> 0 => 1 => 1
[2,2,1]
=> [2,1]
=> 01 => 10 => 1
[4,2]
=> [2]
=> 0 => 1 => 1
[3,2,1]
=> [2,1]
=> 01 => 10 => 1
[2,2,2]
=> [2,2]
=> 00 => 11 => 2
[2,2,1,1]
=> [2,1,1]
=> 011 => 100 => 1
[5,2]
=> [2]
=> 0 => 1 => 1
[4,2,1]
=> [2,1]
=> 01 => 10 => 1
[3,2,2]
=> [2,2]
=> 00 => 11 => 2
[3,2,1,1]
=> [2,1,1]
=> 011 => 100 => 1
[2,2,2,1]
=> [2,2,1]
=> 001 => 110 => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> 0111 => 1000 => 1
[6,2]
=> [2]
=> 0 => 1 => 1
[5,2,1]
=> [2,1]
=> 01 => 10 => 1
[4,4]
=> [4]
=> 0 => 1 => 1
[4,2,2]
=> [2,2]
=> 00 => 11 => 2
[4,2,1,1]
=> [2,1,1]
=> 011 => 100 => 1
[3,3,2]
=> [3,2]
=> 10 => 01 => 1
[3,2,2,1]
=> [2,2,1]
=> 001 => 110 => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> 0111 => 1000 => 1
[2,2,2,2]
=> [2,2,2]
=> 000 => 111 => 3
[2,2,2,1,1]
=> [2,2,1,1]
=> 0011 => 1100 => 1
[7,2]
=> [2]
=> 0 => 1 => 1
[6,2,1]
=> [2,1]
=> 01 => 10 => 1
[5,4]
=> [4]
=> 0 => 1 => 1
[5,2,2]
=> [2,2]
=> 00 => 11 => 2
[5,2,1,1]
=> [2,1,1]
=> 011 => 100 => 1
[4,4,1]
=> [4,1]
=> 01 => 10 => 1
[4,3,2]
=> [3,2]
=> 10 => 01 => 1
[4,2,2,1]
=> [2,2,1]
=> 001 => 110 => 1
[4,2,1,1,1]
=> [2,1,1,1]
=> 0111 => 1000 => 1
[3,3,2,1]
=> [3,2,1]
=> 101 => 010 => 1
[3,2,2,2]
=> [2,2,2]
=> 000 => 111 => 3
[3,2,2,1,1]
=> [2,2,1,1]
=> 0011 => 1100 => 1
[2,2,2,2,1]
=> [2,2,2,1]
=> 0001 => 1110 => 2
[8,2]
=> [2]
=> 0 => 1 => 1
[7,2,1]
=> [2,1]
=> 01 => 10 => 1
[6,4]
=> [4]
=> 0 => 1 => 1
[6,2,2]
=> [2,2]
=> 00 => 11 => 2
[6,2,1,1]
=> [2,1,1]
=> 011 => 100 => 1
[5,4,1]
=> [4,1]
=> 01 => 10 => 1
[5,3,2]
=> [3,2]
=> 10 => 01 => 1
[5,2,2,1]
=> [2,2,1]
=> 001 => 110 => 1
[5,2,1,1,1]
=> [2,1,1,1]
=> 0111 => 1000 => 1
[4,4,2]
=> [4,2]
=> 00 => 11 => 2
[4,4,1,1]
=> [4,1,1]
=> 011 => 100 => 1
[4,3,2,1]
=> [3,2,1]
=> 101 => 010 => 1
[4,2,2,2]
=> [2,2,2]
=> 000 => 111 => 3
[4,2,2,1,1]
=> [2,2,1,1]
=> 0011 => 1100 => 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset. Let $A_n=K[x]/(x^n)$. We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001637
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St001637: Posets ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 60%
Values
[2,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[3,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[4,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[2,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[2,2,1,1]
=> 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[5,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[3,2,2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[2,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,2,1,1,1]
=> 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[6,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[4,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[4,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[4,2,1,1]
=> 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[2,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[2,2,2,1,1]
=> 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[7,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[5,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,2,2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[4,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[4,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[4,2,1,1,1]
=> 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[3,2,2,2]
=> 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 + 1
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[2,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 + 1
[8,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[6,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[6,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,2,1,1]
=> 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[5,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[4,4,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[4,4,1,1]
=> 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[4,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[4,2,2,1,1]
=> 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[3,3,2,2]
=> 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 + 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 + 1
[2,2,2,2,2]
=> 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 4 + 1
[9,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[7,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[7,2,2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[6,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[6,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[6,2,1,1,1]
=> 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[5,4,2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[5,2,2,2]
=> 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 + 1
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[10,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[8,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[8,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,6]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[6,4,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[4,4,4]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[4,4,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[11,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[9,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[7,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[12,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[10,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[10,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[8,6]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[8,4,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[8,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[6,6,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,4,4]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,4,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[4,4,4,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[13,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[11,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[9,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[14,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[12,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[12,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[10,6]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[10,4,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[10,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[8,8]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[8,6,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
Mp00317: Integer partitions odd partsBinary words
Mp00262: Binary words poset of factorsPosets
St001668: Posets ⟶ ℤResult quality: 13% values known / values provided: 13%distinct values known / distinct values provided: 60%
Values
[2,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[3,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[4,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[3,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[2,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[2,2,1,1]
=> 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[5,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[4,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[3,2,2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[3,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[2,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[2,2,1,1,1]
=> 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[6,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[5,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[4,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[4,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[4,2,1,1]
=> 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[3,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[3,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[2,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[2,2,2,1,1]
=> 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[7,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[6,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[5,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[5,2,2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[4,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[4,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[4,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[4,2,1,1,1]
=> 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[3,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[3,2,2,2]
=> 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 + 1
[3,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[2,2,2,2,1]
=> 00001 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 2 + 1
[8,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[7,2,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[6,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[6,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,2,1,1]
=> 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[5,4,1]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[5,3,2]
=> 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[5,2,2,1]
=> 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
[5,2,1,1,1]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[4,4,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[4,4,1,1]
=> 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[4,3,2,1]
=> 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ? = 1 + 1
[4,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[4,2,2,1,1]
=> 00011 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[3,3,2,2]
=> 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,3,2,1,1]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 1 + 1
[3,2,2,2,1]
=> 10001 => ([(0,3),(0,4),(1,2),(1,10),(1,11),(2,8),(2,9),(3,6),(3,7),(4,1),(4,6),(4,7),(6,11),(7,10),(8,5),(9,5),(10,8),(11,9)],12)
=> ? = 2 + 1
[2,2,2,2,2]
=> 00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 4 + 1
[9,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[8,2,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[7,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[7,2,2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[7,2,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[6,4,1]
=> 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[6,3,2]
=> 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[6,2,2,1]
=> 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[6,2,1,1,1]
=> 00111 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[5,4,2]
=> 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2 + 1
[5,4,1,1]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[5,3,2,1]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
[5,2,2,2]
=> 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3 + 1
[5,2,2,1,1]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1 + 1
[10,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[8,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[8,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,6]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[6,4,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[4,4,4]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[4,4,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[11,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[9,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[7,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[12,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[10,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[10,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[8,6]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[8,4,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[8,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[6,6,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,4,4]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[6,4,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[4,4,4,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[13,2]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[11,4]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[9,6]
=> 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[14,2]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[12,4]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[12,2,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[10,6]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[10,4,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[10,2,2,2]
=> 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 3 + 1
[8,8]
=> 00 => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[8,6,2]
=> 000 => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St001207
Mp00202: Integer partitions first row removalInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00201: Dyck paths RingelPermutations
St001207: Permutations ⟶ ℤResult quality: 8% values known / values provided: 8%distinct values known / distinct values provided: 40%
Values
[2,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 1 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 3 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 1 + 1
[7,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 1
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1 + 1
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 1 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 3 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 1 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ? = 2 + 1
[8,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[6,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 1
[5,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1 + 1
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 1 + 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 1
[4,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 2 + 1
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 1 + 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 3 + 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 1 + 1
[3,3,2,2]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 1 + 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => ? = 1 + 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [2,3,6,5,1,4] => ? = 2 + 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [6,3,4,5,1,2] => ? = 4 + 1
[9,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[7,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 1
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 1
[6,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => ? = 1 + 1
[6,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 1
[6,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => ? = 1 + 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 1
[5,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 2 + 1
[5,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,1,2,3,6,4,5] => ? = 1 + 1
[5,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
[5,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 3 + 1
[5,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [2,6,5,1,3,4] => ? = 1 + 1
[4,4,3]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1 + 1
[4,4,2,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,1,2,7,6,3,5] => ? = 1 + 1
[4,4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 1 + 1
[10,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[8,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[11,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[10,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[9,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[12,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[11,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[10,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[13,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[12,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[11,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[14,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[13,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[12,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[15,2]
=> [2]
=> [1,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[14,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 1 + 1
[13,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St001645
Mp00317: Integer partitions odd partsBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001645: Graphs ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 20%
Values
[2,2]
=> 00 => [3] => ([],3)
=> ? = 1 + 5
[3,2]
=> 10 => [1,2] => ([(1,2)],3)
=> ? = 1 + 5
[2,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 5
[4,2]
=> 00 => [3] => ([],3)
=> ? = 1 + 5
[3,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[2,2,2]
=> 000 => [4] => ([],4)
=> ? = 2 + 5
[2,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[5,2]
=> 10 => [1,2] => ([(1,2)],3)
=> ? = 1 + 5
[4,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 5
[3,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> ? = 2 + 5
[3,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[2,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 5
[2,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[6,2]
=> 00 => [3] => ([],3)
=> ? = 1 + 5
[5,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[4,4]
=> 00 => [3] => ([],3)
=> ? = 1 + 5
[4,2,2]
=> 000 => [4] => ([],4)
=> ? = 2 + 5
[4,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[3,3,2]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[3,2,2,1]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[3,2,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[2,2,2,2]
=> 0000 => [5] => ([],5)
=> ? = 3 + 5
[2,2,2,1,1]
=> 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 5
[7,2]
=> 10 => [1,2] => ([(1,2)],3)
=> ? = 1 + 5
[6,2,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 5
[5,4]
=> 10 => [1,2] => ([(1,2)],3)
=> ? = 1 + 5
[5,2,2]
=> 100 => [1,3] => ([(2,3)],4)
=> ? = 2 + 5
[5,2,1,1]
=> 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[4,4,1]
=> 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 5
[4,3,2]
=> 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 5
[4,2,2,1]
=> 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 1 + 5
[4,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[3,3,2,1]
=> 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[3,2,2,2]
=> 1000 => [1,4] => ([(3,4)],5)
=> ? = 3 + 5
[3,2,2,1,1]
=> 10011 => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 5
[2,2,2,2,1]
=> 00001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 5
[8,2]
=> 00 => [3] => ([],3)
=> ? = 1 + 5
[7,2,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[6,4]
=> 00 => [3] => ([],3)
=> ? = 1 + 5
[6,2,2]
=> 000 => [4] => ([],4)
=> ? = 2 + 5
[6,2,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[5,4,1]
=> 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[5,3,2]
=> 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 1 + 5
[5,2,2,1]
=> 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[5,2,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[4,4,2]
=> 000 => [4] => ([],4)
=> ? = 2 + 5
[4,4,1,1]
=> 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[4,3,2,1]
=> 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[4,2,2,2]
=> 0000 => [5] => ([],5)
=> ? = 3 + 5
[4,2,2,1,1]
=> 00011 => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 5
[3,3,2,2]
=> 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1 + 5
[3,3,2,1,1]
=> 11011 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 5
[3,2,2,2,1]
=> 10001 => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 5
[2,2,2,2,2]
=> 00000 => [6] => ([],6)
=> ? = 4 + 5
[6,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[4,4,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[7,2,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[5,4,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[8,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[6,4,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[4,4,3,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[9,2,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[7,4,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[5,4,3,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[10,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[8,4,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[6,6,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[6,4,3,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[4,4,3,3,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[11,2,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[9,4,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[7,6,1,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[7,4,3,1,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[5,4,3,3,1]
=> 10111 => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 1 + 5
[12,2,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[10,4,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[8,6,1,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[8,4,3,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[6,6,3,1,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[6,4,3,3,1]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
[4,4,3,3,3]
=> 00111 => [3,1,1,1] => ([(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)
=> 6 = 1 + 5
Description
The pebbling number of a connected graph.
Matching statistic: St000782
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 20%
Values
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> ? = 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> ? = 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 1
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> ? = 1
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> ? = 3
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> ? = 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> ? = 2
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[6,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 1
[6,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[5,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> ? = 1
[5,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1
[4,4,2]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> ? = 2
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> ? = 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> ? = 3
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> ? = 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> ? = 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> ? = 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> ? = 2
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 4
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[7,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 1
[7,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 2
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[6,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> ? = 1
[6,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> ? = 1
[5,4,2]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> ? = 2
[5,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> ? = 1
[10,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[11,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[12,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[13,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[14,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[13,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
[15,2]
=> [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1
[14,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1
Description
The indicator function of whether a given perfect matching is an L & P matching. An L&P matching is built inductively as follows: starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges. The number of L&P matchings is (see [thm. 1, 2]) $$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001722
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00093: Dyck paths to binary wordBinary words
St001722: Binary words ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 20%
Values
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 1
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 1
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 1
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 3
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[6,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 1
[6,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[5,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
[5,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[4,4,2]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 2
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 3
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => ? = 1
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => ? = 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => ? = 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 2
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 110011110000 => ? = 4
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[7,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 1
[7,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 2
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 1
[6,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
[6,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 1
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 1
[5,4,2]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 2
[5,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 1
[10,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[11,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[12,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[13,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[14,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[13,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
[15,2]
=> [2]
=> [1,1,0,0,1,0]
=> 110010 => 1
[14,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1
Description
The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks. This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length. For example, there are two such chains for the word $0110$: $$ 0110 < 1011 < 1101 < 1110 < 1111 $$ and $$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Mp00202: Integer partitions first row removalInteger partitions
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00082: Standard tableaux to Gelfand-Tsetlin patternGelfand-Tsetlin patterns
St001713: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 20%
Values
[2,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[3,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[4,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 2 + 1
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[5,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 2 + 1
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[6,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 2 + 1
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 3 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[7,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[6,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[5,4]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 2 + 1
[5,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[4,4,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[4,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[4,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[3,3,2,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[3,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 3 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 2 + 1
[8,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[7,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[6,4]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[6,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 2 + 1
[6,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[5,4,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[5,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[5,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[4,4,2]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 1
[4,4,1,1]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[4,3,2,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[4,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 3 + 1
[4,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[2,2,1,1,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[3,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[3,2,2,0,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[3,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[3,2,1,1,0,0,0],[3,2,1,0,0,0],[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 2 + 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[2,2,2,2,0,0,0,0],[2,2,2,1,0,0,0],[2,2,2,0,0,0],[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 4 + 1
[9,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[8,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[7,4]
=> [4]
=> [[1,2,3,4]]
=> [[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[7,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 2 + 1
[7,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[6,4,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[6,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[6,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[6,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
=> ? = 1 + 1
[5,4,2]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [[4,2,0,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 2 + 1
[5,4,1,1]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[4,1,1,0,0,0],[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
=> ? = 1 + 1
[10,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[9,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[11,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[10,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[12,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[11,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[13,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[12,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[14,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[13,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
[15,2]
=> [2]
=> [[1,2]]
=> [[2,0],[1]]
=> 2 = 1 + 1
[14,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> 2 = 1 + 1
Description
The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern.
Matching statistic: St001583
Mp00202: Integer partitions first row removalInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St001583: Permutations ⟶ ℤResult quality: 6% values known / values provided: 6%distinct values known / distinct values provided: 20%
Values
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1 + 2
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 1 + 2
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 1 + 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1 + 2
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 3 + 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 1 + 2
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 1 + 2
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 2
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 1 + 2
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 2
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1 + 2
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 2
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 3 + 2
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 1 + 2
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 2 + 2
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[6,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 1 + 2
[6,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 2
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[5,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 2
[5,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 1 + 2
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 2
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1 + 2
[4,4,2]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 2 + 2
[4,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1 + 2
[4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => ? = 1 + 2
[4,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => ? = 3 + 2
[4,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => ? = 1 + 2
[3,3,2,2]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => ? = 1 + 2
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => ? = 1 + 2
[3,2,2,2,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 2 + 2
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4,1,5,6,7,3] => ? = 4 + 2
[9,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[8,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[7,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 1 + 2
[7,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 2
[7,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 1 + 2
[6,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 2
[6,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 1 + 2
[6,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 1 + 2
[6,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 1 + 2
[5,4,2]
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 2 + 2
[5,4,1,1]
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 1 + 2
[10,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[9,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[11,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[10,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[12,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[11,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[13,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[12,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[14,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[13,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
[15,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 1 + 2
[14,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001198
Mp00321: Integer partitions 2-conjugateInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00030: Dyck paths zeta mapDyck paths
St001198: Dyck paths ⟶ ℤResult quality: 5% values known / values provided: 5%distinct values known / distinct values provided: 20%
Values
[2,2]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[3,2]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[2,2,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[2,2,2]
=> [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]
=> ? = 2 + 1
[2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2 = 1 + 1
[5,2]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[4,2,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[3,2,2]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,2,2,1]
=> [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]
=> ? = 1 + 1
[2,2,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[6,2]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[5,2,1]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,4]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,2,2]
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[4,2,1,1]
=> [6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[3,3,2]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[3,2,2,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,2,1,1,1]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,2,2]
=> [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]
=> ? = 3 + 1
[2,2,2,1,1]
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[7,2]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[6,2,1]
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[5,4]
=> [4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 2 = 1 + 1
[5,2,2]
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[5,2,1,1]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> ? = 1 + 1
[4,4,1]
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[4,3,2]
=> [6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[4,2,2,1]
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[4,2,1,1,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[3,3,2,1]
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[3,2,2,2]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 3 + 1
[3,2,2,1,1]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[2,2,2,2,1]
=> [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]
=> ? = 2 + 1
[8,2]
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 1
[7,2,1]
=> [3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[6,4]
=> [4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[6,2,2]
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[6,2,1,1]
=> [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[5,4,1]
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[5,3,2]
=> [5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[5,2,2,1]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> ? = 1 + 1
[5,2,1,1,1]
=> [4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[4,4,2]
=> [6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> ? = 2 + 1
[4,4,1,1]
=> [6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[4,3,2,1]
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 2 = 1 + 1
[4,2,2,2]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 + 1
[4,2,2,1,1]
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[3,3,2,2]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[3,3,2,1,1]
=> [3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,2,2,2,1]
=> [7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[2,2,2,2,2]
=> [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]
=> ? = 4 + 1
[9,2]
=> [4,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> ? = 1 + 1
[8,2,1]
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[7,4]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0,1,0]
=> ? = 1 + 1
[7,2,2]
=> [6,2,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 2 + 1
[7,2,1,1]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 1 + 1
[6,4,1]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> ? = 1 + 1
[6,3,2]
=> [6,3,2]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[6,2,2,1]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 1
[6,2,1,1,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
[5,4,2]
=> [6,4,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[5,4,1,1]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> ? = 1 + 1
[5,3,2,1]
=> [3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[5,2,2,2]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3 + 1
[5,2,2,1,1]
=> [6,3,1,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[4,4,3]
=> [6,5]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> ? = 1 + 1
[4,4,2,1]
=> [7,4]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 1 + 1
[4,4,1,1,1]
=> [7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ?
=> ? = 1 + 1
[6,6]
=> [4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[3,3,3,2,1]
=> [3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000035The number of left outer peaks of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation.