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Your data matches 69 different statistics following compositions of up to 3 maps.
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Matching statistic: St000124
St000124: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 1
[1,2,3] => 1
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 0
[1,2,3,4] => 1
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 2
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 2
[2,4,3,1] => 0
[3,1,2,4] => 2
[3,1,4,2] => 2
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 2
[3,4,2,1] => 0
[4,1,2,3] => 6
[4,1,3,2] => 0
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 1
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 2
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 1
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 2
[1,3,5,4,2] => 0
[1,4,2,3,5] => 2
[1,4,2,5,3] => 2
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 2
Description
The cardinality of the preimage of the Simion-Schmidt map.
The Simion-Schmidt bijection transforms a [3,1,2]-avoiding permutation into a [3,2,1]-avoiding permutation. More generally, it can be thought of as a map $S$ that turns any permutation into a [3,2,1]-avoiding permutation. This statistic is the size of $S^{-1}(\pi)$ for each permutation $\pi$.
The map $S$ can also be realized using the quotient of the $0$-Hecke Monoid of the symmetric group by the relation $\pi_i \pi_{i+1} \pi_i = \pi_{i+1} \pi_i$, sending each element of the fiber of the quotient to the unique [3,2,1]-avoiding element in that fiber.
Matching statistic: St001570
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Values
[1] => ([],1)
=> ([],1)
=> ? = 1
[1,2] => ([],2)
=> ([],1)
=> ? ∊ {1,1}
[2,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {1,1}
[1,2,3] => ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,2}
[1,3,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2}
[2,1,3] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2}
[2,3,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2}
[3,1,2] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,2}
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,2,3,4] => ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,2,4,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,2,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,4,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,2,3] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,1,3,4] => ([(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,1,4,3] => ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,1,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,1,2,4] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,3,5,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,3,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,3,4] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,3,2,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,5,3,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,5,2,4,3] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,3,2,4] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,4,2,3] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[2,1,3,4,5] => ([(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,3,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,3,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,5,1,4,3] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,3,1,4] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,4,1,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,5,1,4] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,2,1,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,2,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,4,1,2] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,1,3,2,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,3,5,2] => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
[4,2,1,3,5] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 0
Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000175
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000175: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape.
Given a partition $\lambda$ with $r$ parts, the number of semi-standard Young-tableaux of shape $k\lambda$ and boxes with values in $[r]$ grows as a polynomial in $k$. This follows by setting $q=1$ in (7.105) on page 375 of [1], which yields the polynomial
$$p(k) = \prod_{i < j}\frac{k(\lambda_j-\lambda_i)+j-i}{j-i}.$$
The statistic of the degree of this polynomial.
For example, the partition $(3, 2, 1, 1, 1)$ gives
$$p(k) = \frac{-1}{36} (k - 3) (2k - 3) (k - 2)^2 (k - 1)^3$$
which has degree 7 in $k$. Thus, $[3, 2, 1, 1, 1] \mapsto 7$.
This is the same as the number of unordered pairs of different parts, which follows from:
$$\deg p(k)=\sum_{i < j}\begin{cases}1& \lambda_j \neq \lambda_i\\0&\lambda_i=\lambda_j\end{cases}=\sum_{\stackrel{i < j}{\lambda_j \neq \lambda_i}} 1$$
Matching statistic: St000205
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000225
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 78%●distinct values known / distinct values provided: 22%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 78%●distinct values known / distinct values provided: 22%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 78%●distinct values known / distinct values provided: 22%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 22% ●values known / values provided: 78%●distinct values known / distinct values provided: 22%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000749
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St000944
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 78%●distinct values known / distinct values provided: 11%
Values
[1] => [1]
=> []
=> ?
=> ? = 1
[1,2] => [2]
=> []
=> ?
=> ? ∊ {1,1}
[2,1] => [1,1]
=> [1]
=> []
=> ? ∊ {1,1}
[1,2,3] => [3]
=> []
=> ?
=> ? ∊ {1,1,1,1,2}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[2,3,1] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,1,2] => [2,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,2}
[3,2,1] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,3,4] => [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,2,4,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,3,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,1,3] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[2,4,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,1,4,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[3,4,2,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,2,2,2,6}
[4,1,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,1,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4,5] => [5]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,3,5,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,4,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,2,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,2,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,2,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,2,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,3,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,4,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,1,5,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,1,4,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,1,5,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,1,5] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,4,5,1] => [4,1]
=> [1]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,1,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,1,3,5] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,1,5,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,3,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,4,5,1,3] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,4,5,3,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,1,3,4] => [3,2]
=> [2]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,4,6,6,6,6,6,6,6,24}
[2,5,1,4,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[2,5,4,1,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,5,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,5,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,1,4,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,1,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,4,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,5,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,2,5,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,4,2,1,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,2,5,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,4,5,2,1] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[3,5,1,4,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,1,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,2,4,1] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,1,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[3,5,4,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[4,1,3,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[4,1,5,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[4,2,1,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
Description
The 3-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
This stupid comment should not be accepted as an edit!
The following 59 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001498The normalised height of a Nakayama algebra with magnitude 1. St000929The constant term of the character polynomial of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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