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Your data matches 43 different statistics following compositions of up to 3 maps.
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Matching statistic: St001251
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 1
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 2
[2,1,1]
=> 1
[1,1,1,1]
=> 0
[5]
=> 1
[4,1]
=> 0
[3,2]
=> 2
[3,1,1]
=> 1
[2,2,1]
=> 2
[2,1,1,1]
=> 1
[1,1,1,1,1]
=> 0
[6]
=> 1
[5,1]
=> 1
[4,2]
=> 1
[4,1,1]
=> 0
[3,3]
=> 2
[3,2,1]
=> 2
[3,1,1,1]
=> 1
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 1
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 1
[5,2]
=> 2
[5,1,1]
=> 1
[4,3]
=> 1
[4,2,1]
=> 1
[4,1,1,1]
=> 0
[3,3,1]
=> 2
[3,2,2]
=> 3
[3,2,1,1]
=> 2
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 1
[1,1,1,1,1,1,1]
=> 0
[8]
=> 1
[7,1]
=> 0
[6,2]
=> 2
[6,1,1]
=> 1
[5,3]
=> 2
[5,2,1]
=> 2
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001250
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001250: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1]
=> 1
[3]
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1]
=> 1
[1,1,1]
=> [1,1]
=> [2]
=> 1
[4]
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1]
=> 1
[2,2]
=> [2]
=> [1,1]
=> 2
[2,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[5]
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> 1
[3,2]
=> [2]
=> [1,1]
=> 2
[3,1,1]
=> [1,1]
=> [2]
=> 1
[2,2,1]
=> [2,1]
=> [2,1]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[6]
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> 1
[4,2]
=> [2]
=> [1,1]
=> 2
[4,1,1]
=> [1,1]
=> [2]
=> 1
[3,3]
=> [3]
=> [1,1,1]
=> 3
[3,2,1]
=> [2,1]
=> [2,1]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[7]
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1]
=> 1
[5,2]
=> [2]
=> [1,1]
=> 2
[5,1,1]
=> [1,1]
=> [2]
=> 1
[4,3]
=> [3]
=> [1,1,1]
=> 3
[4,2,1]
=> [2,1]
=> [2,1]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 3
[3,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 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]
=> [6]
=> 0
[8]
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [1]
=> 1
[6,2]
=> [2]
=> [1,1]
=> 2
[6,1,1]
=> [1,1]
=> [2]
=> 1
[5,3]
=> [3]
=> [1,1,1]
=> 3
[5,2,1]
=> [2,1]
=> [2,1]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> 4
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 3
[4,2,2]
=> [2,2]
=> [2,2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[3,3,2]
=> [3,2]
=> [2,2,1]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 2
[9]
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> ? = 0
[11]
=> []
=> []
=> ? = 0
[12]
=> []
=> []
=> ? = 0
[13]
=> []
=> []
=> ? = 0
[14]
=> []
=> []
=> ? = 0
[15]
=> []
=> []
=> ? = 0
[16]
=> []
=> []
=> ? = 0
Description
The number of parts of a partition that are not congruent 0 modulo 3.
Matching statistic: St001124
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 56%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 56%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? ∊ {0,1}
[1,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1}
[3]
=> []
=> []
=> []
=> ? ∊ {0,1,1}
[2,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,1}
[4]
=> []
=> []
=> []
=> ? ∊ {0,0,1,1,2}
[3,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,0,1,1,2}
[2,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,0,1,1,2}
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,0,1,1,2}
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,0,1,1,2}
[5]
=> []
=> []
=> []
=> ? ∊ {0,1,1,2,2}
[4,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1,1,2,2}
[3,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,1,2,2}
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,1,2,2}
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,2,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[6]
=> []
=> []
=> []
=> ? ∊ {0,1,1,2,2,3}
[5,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1,1,2,2,3}
[4,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,1,2,2,3}
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,2,2,3}
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[7]
=> []
=> []
=> []
=> ? ∊ {0,1,1,2,2,3}
[6,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1,1,2,2,3}
[5,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,1,2,2,3}
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,2,2,3}
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,2,2,3}
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 3
[8]
=> []
=> []
=> []
=> ? ∊ {0,1,1,2,2,2,3}
[7,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {0,1,1,2,2,2,3}
[6,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {0,1,1,2,2,2,3}
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {0,1,1,2,2,2,3}
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {0,1,1,2,2,2,3}
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,1,1,2,2,2,3}
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,2,2,2,3}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 3
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 3
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> 4
[9]
=> []
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[8,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[7,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 2
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 2
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 2
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 2
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 3
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 3
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 3
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 2
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1]
=> ? ∊ {1,1,1,1,2,2,2,3,3,4}
[10]
=> []
=> []
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,5}
[9,1]
=> [1]
=> [1,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,5}
[8,2]
=> [2]
=> [1,0,1,0]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,5}
[8,1,1]
=> [1,1]
=> [1,1,0,0]
=> []
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,5}
[7,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,1,1,1,2,2,2,2,2,3,3,4,4,5}
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St001491
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 56%
Mp00105: Binary words —complement⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 56%
Values
[1]
=> 1 => 0 => 0 => ? = 0
[2]
=> 0 => 1 => 1 => 1
[1,1]
=> 11 => 00 => 00 => ? = 0
[3]
=> 1 => 0 => 0 => ? ∊ {0,1}
[2,1]
=> 01 => 10 => 01 => 1
[1,1,1]
=> 111 => 000 => 000 => ? ∊ {0,1}
[4]
=> 0 => 1 => 1 => 1
[3,1]
=> 11 => 00 => 00 => ? ∊ {0,0}
[2,2]
=> 00 => 11 => 11 => 2
[2,1,1]
=> 011 => 100 => 001 => 1
[1,1,1,1]
=> 1111 => 0000 => 0000 => ? ∊ {0,0}
[5]
=> 1 => 0 => 0 => ? ∊ {0,0,2}
[4,1]
=> 01 => 10 => 01 => 1
[3,2]
=> 10 => 01 => 10 => 1
[3,1,1]
=> 111 => 000 => 000 => ? ∊ {0,0,2}
[2,2,1]
=> 001 => 110 => 101 => 2
[2,1,1,1]
=> 0111 => 1000 => 0001 => 1
[1,1,1,1,1]
=> 11111 => 00000 => 00000 => ? ∊ {0,0,2}
[6]
=> 0 => 1 => 1 => 1
[5,1]
=> 11 => 00 => 00 => ? ∊ {0,0,1,1,2}
[4,2]
=> 00 => 11 => 11 => 2
[4,1,1]
=> 011 => 100 => 001 => 1
[3,3]
=> 11 => 00 => 00 => ? ∊ {0,0,1,1,2}
[3,2,1]
=> 101 => 010 => 100 => 1
[3,1,1,1]
=> 1111 => 0000 => 0000 => ? ∊ {0,0,1,1,2}
[2,2,2]
=> 000 => 111 => 111 => 3
[2,2,1,1]
=> 0011 => 1100 => 1001 => 2
[2,1,1,1,1]
=> 01111 => 10000 => 00001 => ? ∊ {0,0,1,1,2}
[1,1,1,1,1,1]
=> 111111 => 000000 => 000000 => ? ∊ {0,0,1,1,2}
[7]
=> 1 => 0 => 0 => ? ∊ {0,0,0,2,2,3,3}
[6,1]
=> 01 => 10 => 01 => 1
[5,2]
=> 10 => 01 => 10 => 1
[5,1,1]
=> 111 => 000 => 000 => ? ∊ {0,0,0,2,2,3,3}
[4,3]
=> 01 => 10 => 01 => 1
[4,2,1]
=> 001 => 110 => 101 => 2
[4,1,1,1]
=> 0111 => 1000 => 0001 => 1
[3,3,1]
=> 111 => 000 => 000 => ? ∊ {0,0,0,2,2,3,3}
[3,2,2]
=> 100 => 011 => 110 => 1
[3,2,1,1]
=> 1011 => 0100 => 1000 => 1
[3,1,1,1,1]
=> 11111 => 00000 => 00000 => ? ∊ {0,0,0,2,2,3,3}
[2,2,2,1]
=> 0001 => 1110 => 1101 => 2
[2,2,1,1,1]
=> 00111 => 11000 => 10001 => ? ∊ {0,0,0,2,2,3,3}
[2,1,1,1,1,1]
=> 011111 => 100000 => 000001 => ? ∊ {0,0,0,2,2,3,3}
[1,1,1,1,1,1,1]
=> 1111111 => 0000000 => 0000000 => ? ∊ {0,0,0,2,2,3,3}
[8]
=> 0 => 1 => 1 => 1
[7,1]
=> 11 => 00 => 00 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[6,2]
=> 00 => 11 => 11 => 2
[6,1,1]
=> 011 => 100 => 001 => 1
[5,3]
=> 11 => 00 => 00 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[5,2,1]
=> 101 => 010 => 100 => 1
[5,1,1,1]
=> 1111 => 0000 => 0000 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[4,4]
=> 00 => 11 => 11 => 2
[4,3,1]
=> 011 => 100 => 001 => 1
[4,2,2]
=> 000 => 111 => 111 => 3
[4,2,1,1]
=> 0011 => 1100 => 1001 => 2
[4,1,1,1,1]
=> 01111 => 10000 => 00001 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[3,3,2]
=> 110 => 001 => 010 => 1
[3,3,1,1]
=> 1111 => 0000 => 0000 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[3,2,2,1]
=> 1001 => 0110 => 1100 => 1
[3,2,1,1,1]
=> 10111 => 01000 => 10000 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[3,1,1,1,1,1]
=> 111111 => 000000 => 000000 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[2,2,2,2]
=> 0000 => 1111 => 1111 => 4
[2,2,2,1,1]
=> 00011 => 11100 => 11001 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[2,2,1,1,1,1]
=> 001111 => 110000 => 100001 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[2,1,1,1,1,1,1]
=> 0111111 => 1000000 => 0000001 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[1,1,1,1,1,1,1,1]
=> 11111111 => 00000000 => 00000000 => ? ∊ {0,0,0,0,1,2,2,2,2,3,3}
[9]
=> 1 => 0 => 0 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[8,1]
=> 01 => 10 => 01 => 1
[7,2]
=> 10 => 01 => 10 => 1
[7,1,1]
=> 111 => 000 => 000 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[6,3]
=> 01 => 10 => 01 => 1
[6,2,1]
=> 001 => 110 => 101 => 2
[6,1,1,1]
=> 0111 => 1000 => 0001 => 1
[5,4]
=> 10 => 01 => 10 => 1
[5,3,1]
=> 111 => 000 => 000 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[5,2,2]
=> 100 => 011 => 110 => 1
[5,2,1,1]
=> 1011 => 0100 => 1000 => 1
[5,1,1,1,1]
=> 11111 => 00000 => 00000 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[4,4,1]
=> 001 => 110 => 101 => 2
[4,3,2]
=> 010 => 101 => 011 => 1
[4,3,1,1]
=> 0111 => 1000 => 0001 => 1
[4,2,2,1]
=> 0001 => 1110 => 1101 => 2
[4,2,1,1,1]
=> 00111 => 11000 => 10001 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[4,1,1,1,1,1]
=> 011111 => 100000 => 000001 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[3,3,3]
=> 111 => 000 => 000 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[3,3,2,1]
=> 1101 => 0010 => 0100 => 1
[3,3,1,1,1]
=> 11111 => 00000 => 00000 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[3,2,2,2]
=> 1000 => 0111 => 1110 => 2
[3,2,2,1,1]
=> 10011 => 01100 => 11000 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[3,2,1,1,1,1]
=> 101111 => 010000 => 100000 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[3,1,1,1,1,1,1]
=> 1111111 => 0000000 => 0000000 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[2,2,2,2,1]
=> 00001 => 11110 => 11101 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[2,2,2,1,1,1]
=> 000111 => 111000 => 110001 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[2,2,1,1,1,1,1]
=> 0011111 => 1100000 => 1000001 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[2,1,1,1,1,1,1,1]
=> 01111111 => 10000000 => 00000001 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[1,1,1,1,1,1,1,1,1]
=> 111111111 => 000000000 => 000000000 => ? ∊ {0,0,0,0,2,2,2,2,2,3,3,3,3,3,4,4}
[10]
=> 0 => 1 => 1 => 1
[9,1]
=> 11 => 00 => 00 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
[8,2]
=> 00 => 11 => 11 => 2
[7,3]
=> 11 => 00 => 00 => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,5}
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: St001431
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 56%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001431: Dyck paths ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 56%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? ∊ {0,1}
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1}
[3]
=> []
=> []
=> []
=> ? ∊ {1,1}
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {1,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[4]
=> []
=> []
=> []
=> ? ∊ {0,2}
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,2}
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[5]
=> []
=> []
=> []
=> ? ∊ {0,2}
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,2}
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[6]
=> []
=> []
=> []
=> ? ∊ {2,3}
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {2,3}
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[7]
=> []
=> []
=> []
=> ? ∊ {0,3,3}
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,3,3}
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,3,3}
[8]
=> []
=> []
=> []
=> ? ∊ {0,2,3,3,4}
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,2,3,3,4}
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,2,3,3,4}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,2,3,3,4}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,2,3,3,4}
[9]
=> []
=> []
=> []
=> ? ∊ {0,1,2,3,3,3,3,4,4}
[8,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,1,2,3,3,3,3,4,4}
[7,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,1,2,3,3,3,3,4,4}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,2,3,3,3,3,4,4}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,2,3,3,3,3,4,4}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,1,2,3,3,3,3,4,4}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,2,3,3,3,3,4,4}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,1,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,1,2,3,3,3,3,4,4}
[10]
=> []
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[9,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[1,1,1,1,1,1,1,1,1,1]
=> [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]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3,3,3,3,4,4,5}
[11]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5}
[10,1]
=> [1]
=> [1,0]
=> [1,0]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5}
[5,5,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5}
[5,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5}
[5,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5}
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,5,5}
Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Matching statistic: St001060
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 44%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 44%
Values
[1]
=> 1 => [1] => ([],1)
=> ? = 0
[2]
=> 0 => [1] => ([],1)
=> ? ∊ {0,1}
[1,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,1}
[3]
=> 1 => [1] => ([],1)
=> ? ∊ {0,1,1}
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> ? ∊ {0,1,1}
[1,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,1,1}
[4]
=> 0 => [1] => ([],1)
=> ? ∊ {0,0,1,1,2}
[3,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,1,1,2}
[2,2]
=> 00 => [2] => ([],2)
=> ? ∊ {0,0,1,1,2}
[2,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,2}
[1,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,0,1,1,2}
[5]
=> 1 => [1] => ([],1)
=> ? ∊ {0,0,1,1,1,2}
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,2}
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,2}
[3,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,1,1,1,2}
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,2}
[1,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,2}
[6]
=> 0 => [1] => ([],1)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[5,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[4,2]
=> 00 => [2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[4,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[3,3]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[3,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[2,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[2,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[1,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? ∊ {0,0,1,1,1,1,1,2,2}
[7]
=> 1 => [1] => ([],1)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[5,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[3,3,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[3,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[3,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[3,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,2,1,1,1]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[2,1,1,1,1,1]
=> 011111 => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[1,1,1,1,1,1,1]
=> 1111111 => [7] => ([],7)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2}
[8]
=> 0 => [1] => ([],1)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[7,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[6,2]
=> 00 => [2] => ([],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[6,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[5,3]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[5,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[4,4]
=> 00 => [2] => ([],2)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[4,3,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[4,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[4,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,3,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,3,4}
[2,2,2,1,1]
=> 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[4,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,2,1,1]
=> 10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[6,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,4,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,2,1,1]
=> 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 3
[3,3,2,2]
=> 1100 => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,3,2,1,1]
=> 11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,2,2,2,1,1]
=> 000011 => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 4
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[7,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[6,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[5,4,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[5,2,2,1,1]
=> 10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[4,4,3]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,4,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[4,3,2,2]
=> 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1,1]
=> 01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[4,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[3,3,3,2]
=> 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[3,3,2,2,1]
=> 11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,2,2,1,1]
=> 100011 => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,2,2,2,2,1]
=> 000001 => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[9,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[8,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> 2
Description
The distinguishing index of a graph.
This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism.
If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St000777
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 26% ●values known / values provided: 26%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 2 = 0 + 2
[2]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 3 = 1 + 2
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? = 0 + 2
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 0 + 2
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? = 1 + 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0} + 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,0} + 2
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 1 + 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 1 + 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 2 + 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0} + 2
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,0} + 2
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 1 + 2
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,2} + 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 0 + 2
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,1,2} + 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,1,2} + 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,1,1,2} + 2
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,1,1} + 2
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1} + 2
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1} + 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,1,1} + 2
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 2 + 2
[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] => ([(6,7)],8)
=> ? ∊ {0,0,0,1,1} + 2
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 3 + 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [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)
=> 3 = 1 + 2
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 3 + 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 2 + 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [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)
=> 4 = 2 + 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 2 + 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[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]
=> [1,8] => ([(7,8)],9)
=> ? ∊ {0,0,0,0,1,1,1,2,2,4} + 2
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,2,1] => ([(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 1 + 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 3 + 2
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 1 + 2
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [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)
=> 3 = 1 + 2
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 3 + 2
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 2 + 2
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 1 + 2
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,5,2] => ([(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,7,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,9] => ([(8,9)],10)
=> ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,3,4} + 2
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => ([(0,10),(1,10),(2,10),(3,10),(4,10),(5,10),(6,10),(7,10),(8,10),(9,10)],11)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => ([(0,8),(0,9),(1,8),(1,9),(2,8),(2,9),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1] => ([(0,7),(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,2,1] => ([(0,8),(1,7),(1,8),(2,7),(2,8),(3,7),(3,8),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,3,1] => ([(0,7),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[4,4,1,1]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,6,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,4,5} + 2
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000259
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Values
[1]
=> 1 => [1] => ([],1)
=> 0
[2]
=> 0 => [1] => ([],1)
=> 0
[1,1]
=> 11 => [2] => ([],2)
=> ? = 1
[3]
=> 1 => [1] => ([],1)
=> 0
[2,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[1,1,1]
=> 111 => [3] => ([],3)
=> ? = 1
[4]
=> 0 => [1] => ([],1)
=> 0
[3,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,1,1,2}
[2,2]
=> 00 => [2] => ([],2)
=> ? ∊ {0,1,1,2}
[2,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,1,1,2}
[1,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,1,1,2}
[5]
=> 1 => [1] => ([],1)
=> 0
[4,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[3,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[3,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,1,2}
[2,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,1,2}
[1,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {0,1,2}
[6]
=> 0 => [1] => ([],1)
=> 0
[5,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[4,2]
=> 00 => [2] => ([],2)
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[4,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[3,3]
=> 11 => [2] => ([],2)
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[3,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[2,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[2,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[2,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[1,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? ∊ {0,1,1,1,1,2,2,2,3}
[7]
=> 1 => [1] => ([],1)
=> 0
[6,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[5,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[5,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,1,1,1,2,2,3,3}
[4,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[4,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,2,2,3,3}
[3,3,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,1,1,1,2,2,3,3}
[3,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,2,2,3,3}
[3,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,2,2,3,3}
[3,1,1,1,1]
=> 11111 => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,2,2,3,3}
[2,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,2,1,1,1]
=> 00111 => [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,2,2,3,3}
[2,1,1,1,1,1]
=> 011111 => [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,1,1,1,2,2,3,3}
[1,1,1,1,1,1,1]
=> 1111111 => [7] => ([],7)
=> ? ∊ {0,0,1,1,1,2,2,3,3}
[8]
=> 0 => [1] => ([],1)
=> 0
[7,1]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[6,2]
=> 00 => [2] => ([],2)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[6,1,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[5,3]
=> 11 => [2] => ([],2)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[5,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,1,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[4,4]
=> 00 => [2] => ([],2)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[4,3,1]
=> 011 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[4,2,2]
=> 000 => [3] => ([],3)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[4,2,1,1]
=> 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[4,1,1,1,1]
=> 01111 => [1,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[3,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,3,1,1]
=> 1111 => [4] => ([],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[3,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,1,1]
=> 10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[3,1,1,1,1,1]
=> 111111 => [6] => ([],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[2,2,2,2]
=> 0000 => [4] => ([],4)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[2,2,2,1,1]
=> 00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[2,2,1,1,1,1]
=> 001111 => [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[2,1,1,1,1,1,1]
=> 0111111 => [1,6] => ([(5,6)],7)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[1,1,1,1,1,1,1,1]
=> 11111111 => [8] => ([],8)
=> ? ∊ {0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[9]
=> 1 => [1] => ([],1)
=> 0
[8,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[7,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[7,1,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4}
[6,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[6,1,1,1]
=> 0111 => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4}
[5,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[5,3,1]
=> 111 => [3] => ([],3)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4}
[5,2,2]
=> 100 => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4}
[5,2,1,1]
=> 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,3,3,3,3,3,4,4}
[4,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[4,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2,2,2,1]
=> 00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[10]
=> 0 => [1] => ([],1)
=> 0
[7,2,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,4,1]
=> 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[5,3,2]
=> 110 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[5,2,2,1]
=> 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1]
=> 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,2,2,2,1]
=> 10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[11]
=> 1 => [1] => ([],1)
=> 0
[10,1]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[9,2]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[8,3]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[8,2,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[7,4]
=> 10 => [1,1] => ([(0,1)],2)
=> 1
[6,5]
=> 01 => [1,1] => ([(0,1)],2)
=> 1
[6,4,1]
=> 001 => [2,1] => ([(0,2),(1,2)],3)
=> 2
[6,3,2]
=> 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[6,2,2,1]
=> 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[5,3,2,1]
=> 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001232
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 89%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 89%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 1
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3]
=> []
=> []
=> []
=> ? = 1
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4]
=> []
=> []
=> []
=> ? ∊ {1,2}
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,2}
[5]
=> []
=> []
=> []
=> ? ∊ {1,2,2}
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,2,2}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,2,2}
[6]
=> []
=> []
=> []
=> ? ∊ {1,1,1,2,3}
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {1,1,1,2,3}
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {1,1,1,2,3}
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,2,3}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,1,2,3}
[7]
=> []
=> []
=> []
=> ? ∊ {0,1,1,1,1,2,2,3}
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,1,1,1,1,2,2,3}
[8]
=> []
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[3,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[2,2,1,1,1,1]
=> [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]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[2,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[1,1,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,3,4}
[9]
=> []
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[8,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[7,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[4,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[3,2,1,1,1,1]
=> [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]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[3,1,1,1,1,1,1]
=> [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]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,4}
[9,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[8,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[8,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[7,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001207
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 44%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 44%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? ∊ {0,1}
[1,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1}
[3]
=> []
=> []
=> [] => ? ∊ {1,1}
[2,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[4]
=> []
=> []
=> [] => ? ∊ {0,2}
[3,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,2}
[2,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[5]
=> []
=> []
=> [] => ? ∊ {0,2}
[4,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,2}
[3,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[6]
=> []
=> []
=> [] => ? ∊ {1,2,3}
[5,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,2,3}
[4,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ? ∊ {1,2,3}
[7]
=> []
=> []
=> [] => ? ∊ {0,1,1,3,3}
[6,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1,1,3,3}
[5,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => ? ∊ {0,1,1,3,3}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ? ∊ {0,1,1,3,3}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => ? ∊ {0,1,1,3,3}
[8]
=> []
=> []
=> [] => ? ∊ {0,1,1,2,2,2,3,3,4}
[7,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1,1,2,2,2,3,3,4}
[6,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => ? ∊ {0,1,1,2,2,2,3,3,4}
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => ? ∊ {0,1,1,2,2,2,3,3,4}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ? ∊ {0,1,1,2,2,2,3,3,4}
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => ? ∊ {0,1,1,2,2,2,3,3,4}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => ? ∊ {0,1,1,2,2,2,3,3,4}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => ? ∊ {0,1,1,2,2,2,3,3,4}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6] => ? ∊ {0,1,1,2,2,2,3,3,4}
[9]
=> []
=> []
=> [] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[8,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[7,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[5,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,4,1,5,3,6] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,6] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,6] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,5,2,3,4] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,2,5,3,6] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,1,5,4,7,6] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4,7,6,8] => ? ∊ {0,1,1,1,1,2,2,2,2,2,3,3,3,3,4,4}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[9,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[5,5]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[5,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
[4,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => ? ∊ {0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,5}
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 $K[x]/(x^n)$.
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000260The radius of a connected graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001115The number of even descents of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001964The interval resolution global dimension of a poset. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000624The normalized sum of the minimal distances to a greater element. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001488The number of corners of a skew partition. St001198The 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 $eA$. 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 $eA$. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
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