Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001124
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00312: Integer partitions Glaisher-FranklinInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,1),(0,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [2,2]
=> [2]
=> 0
([(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 0
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: St000159
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00312: Integer partitions Glaisher-FranklinInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [2,2]
=> [2]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 2 = 1 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Matching statistic: St000318
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00312: Integer partitions Glaisher-FranklinInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2 = 0 + 2
([(0,1),(0,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2 = 0 + 2
([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1]
=> 2 = 0 + 2
([(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2 = 0 + 2
([(1,2),(1,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2 = 0 + 2
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2 = 0 + 2
([(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2 = 0 + 2
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,1]
=> 2 = 0 + 2
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [2,2]
=> [2]
=> 2 = 0 + 2
([(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 3 = 1 + 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2 = 0 + 2
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2 = 0 + 2
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2 = 0 + 2
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2 = 0 + 2
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 2 = 0 + 2
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2 = 0 + 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2 = 0 + 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2 = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2 = 0 + 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [2,2,1]
=> [2,1]
=> 3 = 1 + 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2 = 0 + 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [2]
=> 2 = 0 + 2
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2 = 0 + 2
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2 = 0 + 2
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2 = 0 + 2
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2 = 0 + 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 2 = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2 = 0 + 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,1]
=> 2 = 0 + 2
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Matching statistic: St000340
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00312: Integer partitions Glaisher-FranklinInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000340: Dyck paths ⟶ ℤResult quality: 96% values known / values provided: 96%distinct values known / distinct values provided: 100%
Values
([(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
([(0,13),(1,12),(2,13),(2,15),(3,12),(3,15),(5,11),(6,7),(7,4),(8,9),(9,10),(10,7),(11,6),(11,10),(12,8),(13,5),(13,14),(14,9),(14,11),(15,8),(15,14)],16)
=> [7,5,3,1]
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
([(0,11),(1,10),(2,10),(2,13),(3,11),(3,14),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,6),(13,12),(14,7),(14,12)],15)
=> [5,4,3,2,1]
=> [5,3,2,2,1,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> [5,2,2]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> [5,2,2]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,10),(4,5),(4,6),(4,10),(5,9),(5,11),(6,9),(6,11),(7,9),(7,11),(9,8),(10,11),(11,8)],12)
=> [5,3,2,2]
=> [5,3,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> [5,3,2,2,2]
=> [5,3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,6),(1,11),(2,5),(2,11),(3,5),(3,7),(3,11),(4,6),(4,7),(4,11),(5,9),(6,10),(7,9),(7,10),(9,8),(10,8),(11,9),(11,10)],12)
=> [5,3,2,2]
=> [5,3,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> [5,3,2,2,1]
=> [5,3,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(1,5),(1,6),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(5,9),(5,10),(6,9),(6,10),(7,10),(8,9),(8,10),(9,4),(10,4)],11)
=> [5,3,2,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(1,9),(1,10),(2,6),(2,7),(3,5),(3,6),(4,1),(4,5),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> [5,3,2,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,9),(2,10),(3,6),(3,9),(4,7),(4,9),(4,10),(5,1),(5,6),(5,7),(5,10),(6,11),(6,12),(7,11),(7,12),(9,12),(10,11),(10,12),(11,8),(12,8)],13)
=> [5,3,2,2,1]
=> [5,3,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> [5,3,2,2,2]
=> [5,3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> [5,3,2,2,1]
=> [5,3,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,12),(2,7),(2,11),(2,12),(3,7),(3,9),(3,10),(4,6),(4,10),(4,12),(5,6),(5,9),(5,11),(6,14),(7,13),(9,13),(9,14),(10,13),(10,14),(11,13),(11,14),(12,13),(12,14),(13,8),(14,8)],15)
=> [5,3,2,2,2,1]
=> [5,3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 1
([(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)
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
([(0,5),(0,6),(1,4),(1,14),(2,11),(3,10),(4,3),(4,12),(5,1),(5,13),(6,2),(6,13),(8,9),(9,7),(10,7),(11,8),(12,9),(12,10),(13,11),(13,14),(14,8),(14,12)],15)
=> [7,5,3]
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
([(0,5),(0,6),(1,4),(1,15),(2,3),(2,14),(3,8),(4,9),(5,2),(5,13),(6,1),(6,13),(8,10),(9,11),(10,7),(11,7),(12,10),(12,11),(13,14),(13,15),(14,8),(14,12),(15,9),(15,12)],16)
=> [7,5,3,1]
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> [7,4,1]
=> [7,2,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> [9,5,1]
=> [9,5,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
([(0,6),(1,9),(1,10),(2,8),(3,7),(4,3),(4,12),(5,2),(5,12),(6,4),(6,5),(7,9),(7,11),(8,10),(8,11),(9,13),(10,13),(11,13),(12,1),(12,7),(12,8)],14)
=> [7,4,3]
=> [7,3,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
([(0,6),(0,7),(1,9),(2,12),(3,9),(3,12),(4,10),(5,1),(6,5),(7,8),(8,2),(8,3),(9,11),(11,10),(12,4),(12,11)],13)
=> [7,5,1]
=> [7,5,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 1 + 1
([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> [7,5,3]
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> [7,4]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
([(0,7),(1,14),(2,9),(3,10),(4,5),(4,14),(5,6),(5,8),(6,2),(6,11),(7,1),(7,4),(8,10),(8,11),(9,13),(10,12),(11,9),(11,12),(12,13),(14,3),(14,8)],15)
=> [8,5,2]
=> [5,4,4,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,6),(1,9),(2,8),(3,5),(3,7),(4,1),(4,7),(5,2),(5,10),(6,3),(6,4),(7,9),(7,10),(8,12),(9,11),(10,8),(10,11),(11,12)],13)
=> [7,4,2]
=> [7,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,2),(5,1),(5,10),(6,4),(7,8),(8,3),(8,5),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [7,5,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,6),(1,7),(2,8),(3,9),(4,3),(4,7),(5,2),(5,10),(6,1),(6,4),(7,5),(7,9),(9,10),(10,8)],11)
=> [7,4]
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 1
([(0,8),(2,13),(3,11),(4,9),(5,10),(6,3),(6,10),(7,4),(7,12),(8,5),(8,6),(9,13),(10,7),(10,11),(11,12),(12,2),(12,9),(13,1)],14)
=> [9,5]
=> [9,5]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 0 + 1
([(0,6),(0,7),(1,11),(2,9),(3,9),(3,10),(4,5),(5,1),(5,10),(6,4),(7,8),(8,2),(8,3),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [7,5,2]
=> [7,5,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,7),(1,8),(1,9),(2,9),(2,13),(3,8),(3,13),(4,11),(5,10),(6,5),(7,1),(7,2),(7,3),(8,6),(9,12),(11,10),(12,11),(13,4),(13,12)],14)
=> [7,4,3]
=> [7,3,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
([(0,6),(1,12),(2,11),(3,11),(3,12),(4,8),(5,9),(6,1),(6,2),(6,3),(7,8),(7,9),(8,10),(9,10),(11,4),(11,7),(12,5),(12,7)],13)
=> [7,4,2]
=> [7,2,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 1 + 1
([(0,9),(0,10),(1,12),(2,11),(3,11),(3,12),(4,7),(5,8),(6,3),(7,2),(8,1),(9,4),(9,14),(10,5),(10,14),(11,13),(12,13),(14,6)],15)
=> [7,5,3]
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
([(0,7),(0,8),(1,12),(2,11),(3,10),(4,10),(4,11),(5,3),(6,1),(6,13),(7,9),(8,5),(9,2),(9,4),(10,14),(11,6),(11,14),(13,12),(14,13)],15)
=> [8,6,1]
=> [4,4,3,3,1]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0]
=> ? = 2 + 1
([(0,8),(2,11),(2,12),(3,10),(4,9),(5,4),(5,14),(6,3),(6,14),(7,1),(8,5),(8,6),(9,11),(9,13),(10,12),(10,13),(11,15),(12,15),(13,15),(14,2),(14,9),(14,10),(15,7)],16)
=> [9,4,3]
=> [9,3,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 1 + 1
([(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)
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
([(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)
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
([(0,2),(0,3),(1,8),(2,10),(2,11),(3,1),(3,10),(3,11),(5,6),(6,4),(7,4),(8,7),(9,6),(9,7),(10,5),(10,9),(11,5),(11,8),(11,9)],12)
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
([(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)
=> [6,4,2,1]
=> [3,3,2,2,1,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 + 1
([(0,2),(0,3),(1,5),(1,9),(2,10),(2,11),(3,1),(3,10),(3,11),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,6),(11,6),(11,9)],12)
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
([(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)
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> [7,5,3]
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> [7,5,3,1]
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
([(0,3),(0,4),(1,2),(1,14),(2,6),(3,13),(3,15),(4,1),(4,13),(4,15),(6,9),(7,8),(8,10),(9,5),(10,5),(11,8),(11,12),(12,9),(12,10),(13,7),(13,11),(14,6),(14,12),(15,7),(15,11),(15,14)],16)
=> [7,5,3,1]
=> [7,5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
=> ? = 2 + 1
([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> [7,5,3,2]
=> [7,5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
([(0,2),(0,3),(1,11),(1,12),(2,13),(2,14),(3,1),(3,13),(3,14),(5,7),(6,8),(7,4),(8,4),(9,7),(9,8),(10,5),(10,9),(11,6),(11,9),(12,5),(12,6),(13,10),(13,11),(14,10),(14,12)],15)
=> [7,5,3]
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> [7,5,3]
=> [7,5,3]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
([(0,3),(0,4),(1,11),(2,10),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,6),(12,14),(13,7),(13,14),(14,8),(14,9),(15,12),(15,13),(16,10),(16,11),(16,12),(16,13)],17)
=> [7,5,3,2]
=> [7,5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> [7,5,3,2]
=> [7,5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
([(0,3),(0,4),(1,11),(2,12),(2,13),(3,2),(3,15),(3,16),(4,1),(4,15),(4,16),(6,7),(7,9),(8,10),(9,5),(10,5),(11,8),(12,7),(12,14),(13,8),(13,14),(14,9),(14,10),(15,6),(15,12),(16,6),(16,11),(16,13)],17)
=> [7,5,3,2]
=> [7,5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> ? = 2 + 1
Description
The number of non-final maximal constant sub-paths of length greater than one. This is the total number of occurrences of the patterns $110$ and $001$.
Matching statistic: St001001
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St001001: Dyck paths ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1
([(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00093: Dyck paths to binary wordBinary words
St001371: Binary words ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
Description
The length of the longest Yamanouchi prefix of a binary word. This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00093: Dyck paths to binary wordBinary words
St001730: Binary words ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => ? = 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => ? = 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 11001010 => 0
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => ? = 1
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 10111111000000 => ? = 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => ? = 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => ? = 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => ? = 1
([(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => ? = 0
Description
The number of times the path corresponding to a binary word crosses the base line. Interpret each $0$ as a step $(1,-1)$ and $1$ as a step $(1,1)$. Then this statistic counts the number of times the path crosses the $x$-axis.
Matching statistic: St001803
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00033: Dyck paths to two-row standard tableauStandard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 0
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 0
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> ? = 0
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 0
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 0
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> ? = 1
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]]
=> ? = 0
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [[1,3,4,5,6,8],[2,7,9,10,11,12]]
=> ? = 0
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> ? = 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,3,4,5,8],[2,6,7,9,10]]
=> ? = 1
([(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
([(1,4),(4,5),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[1,2,4,5,9],[3,6,7,8,10]]
=> ? = 0
Description
The maximal overlap of the cylindrical tableau associated with a tableau. A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle. The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux. In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck paths
St001195: Dyck paths ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 1 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> ? = 1 + 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 1 + 1
([(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
([(1,4),(4,5),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> ? = 0 + 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Mp00110: Posets Greene-Kleitman invariantInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St001208: Permutations ⟶ ℤResult quality: 7% values known / values provided: 7%distinct values known / distinct values provided: 33%
Values
([(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
([(0,1),(0,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
([(0,2),(1,2)],3)
=> [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 1 = 0 + 1
([(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(1,2),(1,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(0,1),(0,2),(0,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(0,3),(1,3),(2,3)],4)
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 1 = 0 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
([(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 1 = 0 + 1
([(0,3),(2,1),(3,2)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [2,3,4,6,1,5] => ? = 0 + 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
([(2,3),(2,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
([(1,2),(1,3),(1,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(1,3),(1,4),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(1,4),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
([(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => ? = 0 + 1
([(0,4),(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(0,4),(1,2),(1,4),(4,3)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(0,4),(1,2),(1,3)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 1 = 0 + 1
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 1 = 0 + 1
([(1,4),(3,2),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([(0,4),(1,2),(2,4),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([(0,4),(3,2),(4,1),(4,3)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [6,3,4,1,2,5] => ? = 1 + 1
([],6)
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [3,1,4,5,6,7,8,2] => ? = 0 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 + 1
([(3,4),(3,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 + 1
([(2,3),(2,4),(2,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 + 1
([(1,2),(1,3),(1,4),(1,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 + 1
([(0,1),(0,2),(0,3),(0,4),(0,5)],6)
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [7,1,4,5,6,2,3] => ? = 0 + 1
([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(1,3),(1,4),(1,5),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
([(2,3),(2,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(1,4),(1,5),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(2,3),(2,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1 + 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1 + 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1 + 1
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
=> [4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [2,6,4,1,3,5] => ? = 1 + 1
([(3,4),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(2,3),(3,4),(3,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(1,5),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => ? = 1 + 1
([(2,3),(3,5),(5,4)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
([(1,4),(4,5),(5,2),(5,3)],6)
=> [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => ? = 0 + 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001236The dominant dimension of the corresponding Comp-Nakayama algebra.