- FindStat

Your data matches 6 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000525

Mapping

St000525: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],2)
 => 1
([(0,1)],2)
 => 1
([],3)
 => 1
([(1,2)],3)
 => 1
([(0,1),(0,2)],3)
 => 2
([(0,2),(2,1)],3)
 => 1
([(0,2),(1,2)],3)
 => 2
([],4)
 => 1
([(2,3)],4)
 => 1
([(1,2),(1,3)],4)
 => 3
([(0,1),(0,2),(0,3)],4)
 => 3
([(0,2),(0,3),(3,1)],4)
 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
 => 3
([(1,2),(2,3)],4)
 => 1
([(0,3),(3,1),(3,2)],4)
 => 3
([(1,3),(2,3)],4)
 => 3
([(0,3),(1,3),(3,2)],4)
 => 3
([(0,3),(1,3),(2,3)],4)
 => 3
([(0,3),(1,2)],4)
 => 3
([(0,3),(1,2),(1,3)],4)
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
([(0,3),(2,1),(3,2)],4)
 => 1
([(0,3),(1,2),(2,3)],4)
 => 2
([],5)
 => 1
([(3,4)],5)
 => 1
([(2,3),(2,4)],5)
 => 3
([(1,2),(1,3),(1,4)],5)
 => 5
([(0,1),(0,2),(0,3),(0,4)],5)
 => 7
([(0,2),(0,3),(0,4),(4,1)],5)
 => 5
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 8
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 5
([(1,3),(1,4),(4,2)],5)
 => 3
([(0,3),(0,4),(4,1),(4,2)],5)
 => 8
([(1,2),(1,3),(2,4),(3,4)],5)
 => 3
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => 8
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 5
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 3
([(2,3),(3,4)],5)
 => 1
([(1,4),(4,2),(4,3)],5)
 => 3
([(0,4),(4,1),(4,2),(4,3)],5)
 => 5
([(2,4),(3,4)],5)
 => 3
([(1,4),(2,4),(4,3)],5)
 => 3
([(0,4),(1,4),(4,2),(4,3)],5)
 => 3
([(1,4),(2,4),(3,4)],5)
 => 5
([(0,4),(1,4),(2,4),(4,3)],5)
 => 5
([(0,4),(1,4),(2,4),(3,4)],5)
 => 7
([(0,4),(1,4),(2,3)],5)
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => 7
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 2

Description

The number of posets with the same zeta polynomial. The zeta polynomial

$Z$ is the polynomial such that

$Z(m)$ is the number of weakly increasing sequences

$x_1\leq x_2\leq\dots\leq x_{m−1}$ of elements of the poset. See section 3.12 of [1]. Since

$Z(q) = \sum_{k\geq 1} \binom{q-2}{k-1} c_k,$ where

$c_k$ is the number of chains of length

$k$ , this statistic is the same as the number of posets with the same chain polynomial.

Matching statistic: St001880

Mapping

Mp00206: Posets —antichains of maximal size⟶ Lattices
Mp00263: Lattices —join irreducibles⟶ Posets
St001880: Posets ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 21%

Values

([],2)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1}
([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1}
([],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,2,2}
([(1,2)],3)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,2,2}
([(0,1),(0,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,2,2}
([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,2,2}
([(0,2),(1,2)],3)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,2,2}
([],4)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(1,2),(1,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,2,2,3,3,3,3,3,3,3,3}
([],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(3,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,1),(0,2),(0,3),(0,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,2),(0,3),(0,4),(4,1)],5)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(0,4),(4,1),(4,2)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,2),(1,3),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(0,4),(3,2),(4,1)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],2)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => ([(0,1)],2)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,4),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(4,1),(4,2),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(4,2),(4,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ([],1)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ([],0)
 => ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(1,2),(2,3),(2,4)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,3),(0,4),(3,5),(4,1),(4,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(2,3),(3,5),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(1,5),(2,3),(2,5),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,3),(1,4),(2,4),(2,5),(3,1),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(1,4),(1,5),(2,3),(2,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(0,5),(1,2),(1,4),(2,5),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,2),(0,5),(1,4),(1,5),(2,4),(4,3),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,5),(1,4),(1,5),(2,3),(2,4),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(0,5),(1,2),(1,4),(2,3),(3,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
([(0,2),(0,4),(2,5),(3,1),(3,5),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(0,5),(1,3),(3,4),(3,5),(5,2)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(0,5),(1,2),(2,3),(2,5),(3,4)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(1,5),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,5),(1,4),(4,2),(4,5),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,4),(0,5),(1,2),(2,3),(3,4),(3,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,5),(1,2),(2,3),(2,5),(3,4),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3
([(0,5),(1,3),(3,4),(4,2),(4,5)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 5
([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,2),(2,1)],3)
 => 3

Description

The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.

Matching statistic: St001695

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 36%

Values

([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 0 = 1 - 1
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
 => ? ∊ {1,2,3,3,3,3,3} - 1
([(2,3)],4)
 => [6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => ? ∊ {1,2,3,3,3,3,3} - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [[1,2,7,8,9,10],[3,4],[5,6]]
 => ? ∊ {1,2,3,3,3,3,3} - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => ? ∊ {1,2,3,3,3,3,3} - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 2 = 3 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [[1,2,7,8,9,10],[3,4],[5,6]]
 => ? ∊ {1,2,3,3,3,3,3} - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => ? ∊ {1,2,3,3,3,3,3} - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => ? ∊ {1,2,3,3,3,3,3} - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 1 = 2 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(3,4)],5)
 => [6,6,6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [[1,2,15,16,17,18],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [[1,2,17],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 4 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 2 = 3 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [[1,2,15,16,17,18],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [[1,2,17],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [[1,2,5,9,10,11,12,13],[3,4,8],[6,7]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [[1,2,3,4,5,6,13,14,15,16],[7,8,9,10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [[1,2,3,7,8,9,10,11],[4,5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [[1,2,11,12,13,14],[3,4],[5,6],[7,8],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [[1,2,5,6,7,8,9],[3,4]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 2 = 3 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [[1,2,5,6,7,8,9,10,11,12],[3,4]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [[1,2,5,9,10,11,12,13],[3,4,8],[6,7]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 4 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 0 = 1 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 0 = 1 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 4 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1

Description

The natural comajor index of a standard Young tableau. A natural descent of a standard tableau

$T$ is an entry

$i$ such that

$i+1$ appears in a higher row than

$i$ in English notation. The natural comajor index of a tableau of size

$n$ with natural descent set

$D$ is then

$\sum_{d\in D} n-d$ .

Matching statistic: St001698

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 36%

Values

([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 0 = 1 - 1
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(2,3)],4)
 => [6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [[1,2,7,8,9,10],[3,4],[5,6]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 2 = 3 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [[1,2,7,8,9,10],[3,4],[5,6]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [[1,2,9],[3,4],[5,6],[7,8]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(3,4)],5)
 => [6,6,6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [[1,2,15,16,17,18],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [[1,2,17],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 4 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [[1,2,3,10],[4,5,6],[7,8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 4 = 5 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 4 = 5 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [[1,2,15,16,17,18],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,9,10],[3,4],[5,6],[7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [[1,2,17],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [[1,2,5,9,10,11,12,13],[3,4,8],[6,7]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [[1,2,3,4,5,6,13,14,15,16],[7,8,9,10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [[1,2,3,7,8,9,10,11],[4,5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [[1,2,11,12,13,14],[3,4],[5,6],[7,8],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [[1,2,5,6,7,8,9],[3,4]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 4 = 5 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [[1,2,3,4,9],[5,6,7,8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [[1,2,5,6,7,8,9,10,11,12],[3,4]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [[1,2,7,8,9,10,11],[3,4],[5,6]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [[1,2,5,9,10,11,12,13],[3,4,8],[6,7]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 4 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 0 = 1 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 0 = 1 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 4 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [[1,2,5,6,7,8],[3,4]]
 => 4 = 5 - 1

Description

The comajor index of a standard tableau minus the weighted size of its shape.

Matching statistic: St001699

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 36%

Values

([],2)
 => [2,2]
 => [[1,2],[3,4]]
 => 0 = 1 - 1
([(0,1)],2)
 => [3]
 => [[1,2,3]]
 => 0 = 1 - 1
([],3)
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 0 = 1 - 1
([(1,2)],3)
 => [6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
([(0,1),(0,2)],3)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 1 = 2 - 1
([(0,2),(2,1)],3)
 => [4]
 => [[1,2,3,4]]
 => 0 = 1 - 1
([(0,2),(1,2)],3)
 => [3,2]
 => [[1,2,3],[4,5]]
 => 1 = 2 - 1
([],4)
 => [2,2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(2,3)],4)
 => [6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(1,2),(1,3)],4)
 => [6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 2 = 3 - 1
([(1,2),(2,3)],4)
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 0 = 1 - 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 2 = 3 - 1
([(1,3),(2,3)],4)
 => [6,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [[1,2,3,4],[5,6]]
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,3),(1,2)],4)
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => ? ∊ {1,2,2,3,3,3,3} - 1
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => 2 = 3 - 1
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [[1,2,3,4,5]]
 => 0 = 1 - 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20],[21,22],[23,24],[25,26],[27,28],[29,30],[31,32]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(3,4)],5)
 => [6,6,6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12,13]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [[1,2,3,4,5,6,7,8,9,10,11,12,13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 3 = 4 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [[1,2,3,4],[5,6,7],[8,9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => 4 = 5 - 1
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14],[15,16],[17,18],[19,20]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [[1,2,3,4],[5,6,7,8],[9,10],[11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => 4 = 5 - 1
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [[1,2,3,4],[5,6],[7,8],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12,13],[14,15],[16,17]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [[1,2,3,4,5,6,7],[8,9,10,11,12,13]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(2,3)],5)
 => [6,6,6]
 => [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [[1,2,3,4,5,6,7,8],[9,10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [[1,2,3,4,5,6],[7,8],[9,10],[11,12],[13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [[1,2,3,4,5,6,7],[8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [[1,2,3,4],[5,6],[7,8]]
 => 4 = 5 - 1
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [[1,2,3,4,5,6,7,8,9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14,15]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [[1,2,3,4,5],[6,7,8,9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [[1,2,3,4,5,6,7,8,9,10],[11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 0 = 1 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [[1,2,3,4,5,6,7],[8,9],[10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [[1,2,3,4,5,6,7,8],[9,10,11],[12,13]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [[1,2,3,4,5],[6,7,8],[9,10],[11,12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8} - 1
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 3 = 4 - 1
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 0 = 1 - 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [[1,2,3,4,5,6,7,8]]
 => 0 = 1 - 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [[1,2,3,4,5,6]]
 => 0 = 1 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [[1,2,3,4,5],[6,7]]
 => 3 = 4 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => 4 = 5 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => 4 = 5 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => 4 = 5 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => 4 = 5 - 1
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 0 = 1 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => 4 = 5 - 1

Description

The major index of a standard tableau minus the weighted size of its shape.

Matching statistic: St001804

Mapping

Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001804: Standard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 36%

Values

([],2)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 1
([(0,1)],2)
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([],3)
 => [2,2,2,2]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 1
([(1,2)],3)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 1
([(0,1),(0,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
([(0,2),(2,1)],3)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(0,2),(1,2)],3)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 2
([],4)
 => [2,2,2,2,2,2,2,2]
 => [8,8]
 => [[1,2,3,4,5,6,7,8],[9,10,11,12,13,14,15,16]]
 => ? ∊ {1,2,3,3,3,3,3}
([(2,3)],4)
 => [6,6]
 => [2,2,2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
 => ? ∊ {1,2,3,3,3,3,3}
([(1,2),(1,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => [[1,6,7],[2,9,10],[3],[4],[5],[8]]
 => ? ∊ {1,2,3,3,3,3,3}
([(0,1),(0,2),(0,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => ? ∊ {1,2,3,3,3,3,3}
([(0,2),(0,3),(3,1)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 1
([(0,1),(0,2),(1,3),(2,3)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3
([(1,2),(2,3)],4)
 => [4,4]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 1
([(0,3),(3,1),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3
([(1,3),(2,3)],4)
 => [6,2,2]
 => [3,3,1,1,1,1]
 => [[1,6,7],[2,9,10],[3],[4],[5],[8]]
 => ? ∊ {1,2,3,3,3,3,3}
([(0,3),(1,3),(3,2)],4)
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [3,2,2,2]
 => [4,4,1]
 => [[1,3,4,5],[2,7,8,9],[6]]
 => ? ∊ {1,2,3,3,3,3,3}
([(0,3),(1,2)],4)
 => [3,3,3]
 => [3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9]]
 => ? ∊ {1,2,3,3,3,3,3}
([(0,3),(1,2),(1,3)],4)
 => [5,3]
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 2
([(0,3),(2,1),(3,2)],4)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(0,3),(1,2),(2,3)],4)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 1
([],5)
 => [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]
 => [16,16]
 => [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],[17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(3,4)],5)
 => [6,6,6,6]
 => [4,4,4,4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20],[21,22,23,24]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(2,3),(2,4)],5)
 => [6,6,2,2,2,2]
 => [6,6,2,2,2,2]
 => [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,2),(1,3),(1,4)],5)
 => [6,2,2,2,2,2,2]
 => [7,7,1,1,1,1]
 => [[1,6,7,8,9,10,11],[2,13,14,15,16,17,18],[3],[4],[5],[12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,1),(0,2),(0,3),(0,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [8,8,1]
 => [[1,3,4,5,6,7,8,9],[2,11,12,13,14,15,16,17],[10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,2),(0,3),(0,4),(4,1)],5)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,3),(1,4),(4,2)],5)
 => [14]
 => [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[14]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(0,4),(4,1),(4,2)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,2),(1,3),(2,4),(3,4)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 4
([(0,3),(0,4),(3,2),(4,1)],5)
 => [4,3,3]
 => [3,3,3,1]
 => [[1,3,4],[2,6,7],[5,9,10],[8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 3
([(2,3),(3,4)],5)
 => [4,4,4,4]
 => [4,4,4,4]
 => [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,4),(4,2),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(4,1),(4,2),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(2,4),(3,4)],5)
 => [6,6,2,2,2,2]
 => [6,6,2,2,2,2]
 => [[1,2,11,12,13,14],[3,4,17,18,19,20],[5,6],[7,8],[9,10],[15,16]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,4),(2,4),(4,3)],5)
 => [4,4,2,2]
 => [4,4,2,2]
 => [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(4,2),(4,3)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [6,2,2,2,2,2,2]
 => [7,7,1,1,1,1]
 => [[1,6,7,8,9,10,11],[2,13,14,15,16,17,18],[3],[4],[5],[12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(2,4),(4,3)],5)
 => [4,2,2,2]
 => [4,4,1,1]
 => [[1,4,5,6],[2,8,9,10],[3],[7]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(2,4),(3,4)],5)
 => [3,2,2,2,2,2,2,2]
 => [8,8,1]
 => [[1,3,4,5,6,7,8,9],[2,11,12,13,14,15,16,17],[10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(2,3)],5)
 => [6,3,3,3]
 => [4,4,4,1,1,1]
 => [[1,5,6,7],[2,9,10,11],[3,13,14,15],[4],[8],[12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,3),(2,3),(2,4)],5)
 => [8,3,2]
 => [3,3,2,1,1,1,1,1]
 => [[1,7,10],[2,9,13],[3,12],[4],[5],[6],[8],[11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5,3,2,2]
 => [4,4,2,1,1]
 => [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [3,2,2,2,2]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(2,3),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 4
([(0,4),(1,3),(2,3),(3,4)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(2,3),(2,4)],5)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,4),(2,3),(3,4)],5)
 => [7,6]
 => [2,2,2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,4),(2,3)],5)
 => [6,6,6]
 => [3,3,3,3,3,3]
 => [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,4),(2,3),(2,4)],5)
 => [10,6]
 => [2,2,2,2,2,2,1,1,1,1]
 => [[1,6],[2,8],[3,10],[4,12],[5,14],[7,16],[9],[11],[13],[15]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,2),(1,4),(2,3)],5)
 => [8,3]
 => [2,2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(1,3),(1,4),(2,3),(2,4)],5)
 => [6,2,2,2,2]
 => [5,5,1,1,1,1]
 => [[1,6,7,8,9],[2,11,12,13,14],[3],[4],[5],[10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
 => [7,2]
 => [2,2,1,1,1,1,1]
 => [[1,7],[2,9],[3],[4],[5],[6],[8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
 => [4,2,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 3
([(0,4),(1,2),(1,4),(4,3)],5)
 => [10]
 => [1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,2),(1,3)],5)
 => [6,3,3,3]
 => [4,4,4,1,1,1]
 => [[1,5,6,7],[2,9,10,11],[3,13,14,15],[4],[8],[12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,2),(1,3),(1,4)],5)
 => [6,5,3]
 => [3,3,3,2,2,1]
 => [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,2),(0,4),(3,1),(4,3)],5)
 => [5,4]
 => [2,2,2,2,1]
 => [[1,3],[2,5],[4,7],[6,9],[8]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,4),(1,2),(1,3),(3,4)],5)
 => [10,2]
 => [2,2,1,1,1,1,1,1,1,1]
 => [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [7,2,2]
 => [3,3,1,1,1,1,1]
 => [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(0,4),(1,2),(1,4)],5)
 => [8,3,2]
 => [3,3,2,1,1,1,1,1]
 => [[1,7,10],[2,9,13],[3,12],[4],[5],[6],[8],[11]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [5,3,2,2]
 => [4,4,2,1,1]
 => [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
 => [3,2,2,2,2]
 => [5,5,1]
 => [[1,3,4,5,6],[2,8,9,10,11],[7]]
 => ? ∊ {1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8}
([(0,3),(3,4),(4,1),(4,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 4
([(0,4),(1,2),(2,4),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 1
([(0,4),(3,2),(4,1),(4,3)],5)
 => [8]
 => [1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8]]
 => 1
([(0,4),(2,3),(3,1),(4,2)],5)
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 4
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 5
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 5
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 5
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 5
([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => [6,2]
 => [2,2,1,1,1,1]
 => [[1,6],[2,8],[3],[4],[5],[7]]
 => 5

Description

The minimal height of the rectangular inner shape in a cylindrical tableau associated to 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. This statistic equals

$\max_C\big(\ell(C) - \ell(T)\big)$ , where

$\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.