searching the database
Your data matches 34 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000165
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000165: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[1,2] => 3
[2,1] => 3
[1,1,1] => 3
[1,1,2] => 4
[1,2,1] => 4
[2,1,1] => 4
[1,1,3] => 5
[1,3,1] => 5
[3,1,1] => 5
[1,2,2] => 5
[2,1,2] => 5
[2,2,1] => 5
[1,2,3] => 6
[1,3,2] => 6
[2,1,3] => 6
[2,3,1] => 6
[3,1,2] => 6
[3,2,1] => 6
[1,1,1,1] => 4
[1,1,1,2] => 5
[1,1,2,1] => 5
[1,2,1,1] => 5
[2,1,1,1] => 5
[1,1,1,3] => 6
[1,1,3,1] => 6
[1,3,1,1] => 6
[3,1,1,1] => 6
[1,1,1,4] => 7
[1,1,4,1] => 7
[1,4,1,1] => 7
[4,1,1,1] => 7
[1,1,2,2] => 6
[1,2,1,2] => 6
[1,2,2,1] => 6
[2,1,1,2] => 6
[2,1,2,1] => 6
[2,2,1,1] => 6
[1,1,2,3] => 7
[1,1,3,2] => 7
[1,2,1,3] => 7
[1,2,3,1] => 7
[1,3,1,2] => 7
[1,3,2,1] => 7
[2,1,1,3] => 7
[2,1,3,1] => 7
[2,3,1,1] => 7
[3,1,1,2] => 7
[3,1,2,1] => 7
Description
The sum of the entries of a parking function.
The generating function for parking functions by sum is the evaluation at $x=1$ and $y=1/q$ of the Tutte polynomial of the complete graph, multiplied by $q^\binom{n}{2}$.
Matching statistic: St000103
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00302: Parking functions —insertion tableau⟶ Semistandard tableaux
St000103: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000103: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> 1
[1,1] => [[1,1]]
=> 2
[1,2] => [[1,2]]
=> 3
[2,1] => [[1],[2]]
=> 3
[1,1,1] => [[1,1,1]]
=> 3
[1,1,2] => [[1,1,2]]
=> 4
[1,2,1] => [[1,1],[2]]
=> 4
[2,1,1] => [[1,1],[2]]
=> 4
[1,1,3] => [[1,1,3]]
=> 5
[1,3,1] => [[1,1],[3]]
=> 5
[3,1,1] => [[1,1],[3]]
=> 5
[1,2,2] => [[1,2,2]]
=> 5
[2,1,2] => [[1,2],[2]]
=> 5
[2,2,1] => [[1,2],[2]]
=> 5
[1,2,3] => [[1,2,3]]
=> 6
[1,3,2] => [[1,2],[3]]
=> 6
[2,1,3] => [[1,3],[2]]
=> 6
[2,3,1] => [[1,3],[2]]
=> 6
[3,1,2] => [[1,2],[3]]
=> 6
[3,2,1] => [[1],[2],[3]]
=> 6
[1,1,1,1] => [[1,1,1,1]]
=> 4
[1,1,1,2] => [[1,1,1,2]]
=> 5
[1,1,2,1] => [[1,1,1],[2]]
=> 5
[1,2,1,1] => [[1,1,1],[2]]
=> 5
[2,1,1,1] => [[1,1,1],[2]]
=> 5
[1,1,1,3] => [[1,1,1,3]]
=> 6
[1,1,3,1] => [[1,1,1],[3]]
=> 6
[1,3,1,1] => [[1,1,1],[3]]
=> 6
[3,1,1,1] => [[1,1,1],[3]]
=> 6
[1,1,1,4] => [[1,1,1,4]]
=> 7
[1,1,4,1] => [[1,1,1],[4]]
=> 7
[1,4,1,1] => [[1,1,1],[4]]
=> 7
[4,1,1,1] => [[1,1,1],[4]]
=> 7
[1,1,2,2] => [[1,1,2,2]]
=> 6
[1,2,1,2] => [[1,1,2],[2]]
=> 6
[1,2,2,1] => [[1,1,2],[2]]
=> 6
[2,1,1,2] => [[1,1,2],[2]]
=> 6
[2,1,2,1] => [[1,1],[2,2]]
=> 6
[2,2,1,1] => [[1,1],[2,2]]
=> 6
[1,1,2,3] => [[1,1,2,3]]
=> 7
[1,1,3,2] => [[1,1,2],[3]]
=> 7
[1,2,1,3] => [[1,1,3],[2]]
=> 7
[1,2,3,1] => [[1,1,3],[2]]
=> 7
[1,3,1,2] => [[1,1,2],[3]]
=> 7
[1,3,2,1] => [[1,1],[2],[3]]
=> 7
[2,1,1,3] => [[1,1,3],[2]]
=> 7
[2,1,3,1] => [[1,1],[2,3]]
=> 7
[2,3,1,1] => [[1,1],[2,3]]
=> 7
[3,1,1,2] => [[1,1,2],[3]]
=> 7
[3,1,2,1] => [[1,1],[2],[3]]
=> 7
Description
The sum of the entries of a semistandard tableau.
Matching statistic: St000639
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 93% ●values known / values provided: 100%●distinct values known / distinct values provided: 93%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000639: Posets ⟶ ℤResult quality: 93% ●values known / values provided: 100%●distinct values known / distinct values provided: 93%
Values
[1] => [1,0]
=> ([],1)
=> ? = 1
[1,1] => [1,1,0,0]
=> ([],2)
=> 2
[1,2] => [1,0,1,0]
=> ([(0,1)],2)
=> 3
[2,1] => [1,0,1,0]
=> ([(0,1)],2)
=> 3
[1,1,1] => [1,1,1,0,0,0]
=> ([],3)
=> 3
[1,1,2] => [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 4
[1,2,1] => [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 4
[2,1,1] => [1,1,0,1,0,0]
=> ([(1,2)],3)
=> 4
[1,1,3] => [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 5
[1,3,1] => [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 5
[3,1,1] => [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 5
[1,2,2] => [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 5
[2,1,2] => [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 5
[2,2,1] => [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 5
[1,2,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 6
[1,3,2] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 6
[2,1,3] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 6
[2,3,1] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 6
[3,1,2] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 6
[3,2,1] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 6
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 5
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 5
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 5
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> 5
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 6
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 6
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 6
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> 6
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 7
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 7
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 7
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 7
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 6
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 6
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 6
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 6
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 6
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 6
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
[3,2,1,1] => [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 7
Description
The number of relations in a poset.
This is the number of intervals $x,y$ with $x\leq y$ in the poset, and therefore the dimension of the posets incidence algebra.
Matching statistic: St000114
Mp00302: Parking functions —insertion tableau⟶ Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00078: Gelfand-Tsetlin patterns —Schuetzenberger involution⟶ Gelfand-Tsetlin patterns
St000114: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 93%
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00078: Gelfand-Tsetlin patterns —Schuetzenberger involution⟶ Gelfand-Tsetlin patterns
St000114: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 93%
Values
[1] => [[1]]
=> [[1]]
=> [[1]]
=> 1
[1,1] => [[1,1]]
=> [[2]]
=> [[2]]
=> 2
[1,2] => [[1,2]]
=> [[2,0],[1]]
=> [[2,0],[1]]
=> 3
[2,1] => [[1],[2]]
=> [[1,1],[1]]
=> [[1,1],[1]]
=> 3
[1,1,1] => [[1,1,1]]
=> [[3]]
=> [[3]]
=> 3
[1,1,2] => [[1,1,2]]
=> [[3,0],[2]]
=> [[3,0],[1]]
=> 4
[1,2,1] => [[1,1],[2]]
=> [[2,1],[2]]
=> [[2,1],[1]]
=> 4
[2,1,1] => [[1,1],[2]]
=> [[2,1],[2]]
=> [[2,1],[1]]
=> 4
[1,1,3] => [[1,1,3]]
=> [[3,0,0],[2,0],[2]]
=> [[3,0,0],[1,0],[1]]
=> 5
[1,3,1] => [[1,1],[3]]
=> [[2,1,0],[2,0],[2]]
=> [[2,1,0],[1,0],[1]]
=> 5
[3,1,1] => [[1,1],[3]]
=> [[2,1,0],[2,0],[2]]
=> [[2,1,0],[1,0],[1]]
=> 5
[1,2,2] => [[1,2,2]]
=> [[3,0],[1]]
=> [[3,0],[2]]
=> 5
[2,1,2] => [[1,2],[2]]
=> [[2,1],[1]]
=> [[2,1],[2]]
=> 5
[2,2,1] => [[1,2],[2]]
=> [[2,1],[1]]
=> [[2,1],[2]]
=> 5
[1,2,3] => [[1,2,3]]
=> [[3,0,0],[2,0],[1]]
=> [[3,0,0],[2,0],[1]]
=> 6
[1,3,2] => [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> [[2,1,0],[1,1],[1]]
=> 6
[2,1,3] => [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> [[2,1,0],[2,0],[1]]
=> 6
[2,3,1] => [[1,3],[2]]
=> [[2,1,0],[1,1],[1]]
=> [[2,1,0],[2,0],[1]]
=> 6
[3,1,2] => [[1,2],[3]]
=> [[2,1,0],[2,0],[1]]
=> [[2,1,0],[1,1],[1]]
=> 6
[3,2,1] => [[1],[2],[3]]
=> [[1,1,1],[1,1],[1]]
=> [[1,1,1],[1,1],[1]]
=> 6
[1,1,1,1] => [[1,1,1,1]]
=> [[4]]
=> [[4]]
=> 4
[1,1,1,2] => [[1,1,1,2]]
=> [[4,0],[3]]
=> [[4,0],[1]]
=> 5
[1,1,2,1] => [[1,1,1],[2]]
=> [[3,1],[3]]
=> [[3,1],[1]]
=> 5
[1,2,1,1] => [[1,1,1],[2]]
=> [[3,1],[3]]
=> [[3,1],[1]]
=> 5
[2,1,1,1] => [[1,1,1],[2]]
=> [[3,1],[3]]
=> [[3,1],[1]]
=> 5
[1,1,1,3] => [[1,1,1,3]]
=> [[4,0,0],[3,0],[3]]
=> [[4,0,0],[1,0],[1]]
=> 6
[1,1,3,1] => [[1,1,1],[3]]
=> [[3,1,0],[3,0],[3]]
=> [[3,1,0],[1,0],[1]]
=> 6
[1,3,1,1] => [[1,1,1],[3]]
=> [[3,1,0],[3,0],[3]]
=> [[3,1,0],[1,0],[1]]
=> 6
[3,1,1,1] => [[1,1,1],[3]]
=> [[3,1,0],[3,0],[3]]
=> [[3,1,0],[1,0],[1]]
=> 6
[1,1,1,4] => [[1,1,1,4]]
=> [[4,0,0,0],[3,0,0],[3,0],[3]]
=> [[4,0,0,0],[1,0,0],[1,0],[1]]
=> 7
[1,1,4,1] => [[1,1,1],[4]]
=> [[3,1,0,0],[3,0,0],[3,0],[3]]
=> [[3,1,0,0],[1,0,0],[1,0],[1]]
=> 7
[1,4,1,1] => [[1,1,1],[4]]
=> [[3,1,0,0],[3,0,0],[3,0],[3]]
=> [[3,1,0,0],[1,0,0],[1,0],[1]]
=> 7
[4,1,1,1] => [[1,1,1],[4]]
=> [[3,1,0,0],[3,0,0],[3,0],[3]]
=> [[3,1,0,0],[1,0,0],[1,0],[1]]
=> 7
[1,1,2,2] => [[1,1,2,2]]
=> [[4,0],[2]]
=> [[4,0],[2]]
=> 6
[1,2,1,2] => [[1,1,2],[2]]
=> [[3,1],[2]]
=> [[3,1],[2]]
=> 6
[1,2,2,1] => [[1,1,2],[2]]
=> [[3,1],[2]]
=> [[3,1],[2]]
=> 6
[2,1,1,2] => [[1,1,2],[2]]
=> [[3,1],[2]]
=> [[3,1],[2]]
=> 6
[2,1,2,1] => [[1,1],[2,2]]
=> [[2,2],[2]]
=> [[2,2],[2]]
=> 6
[2,2,1,1] => [[1,1],[2,2]]
=> [[2,2],[2]]
=> [[2,2],[2]]
=> 6
[1,1,2,3] => [[1,1,2,3]]
=> [[4,0,0],[3,0],[2]]
=> [[4,0,0],[2,0],[1]]
=> 7
[1,1,3,2] => [[1,1,2],[3]]
=> [[3,1,0],[3,0],[2]]
=> [[3,1,0],[1,1],[1]]
=> 7
[1,2,1,3] => [[1,1,3],[2]]
=> [[3,1,0],[2,1],[2]]
=> [[3,1,0],[2,0],[1]]
=> 7
[1,2,3,1] => [[1,1,3],[2]]
=> [[3,1,0],[2,1],[2]]
=> [[3,1,0],[2,0],[1]]
=> 7
[1,3,1,2] => [[1,1,2],[3]]
=> [[3,1,0],[3,0],[2]]
=> [[3,1,0],[1,1],[1]]
=> 7
[1,3,2,1] => [[1,1],[2],[3]]
=> [[2,1,1],[2,1],[2]]
=> [[2,1,1],[1,1],[1]]
=> 7
[2,1,1,3] => [[1,1,3],[2]]
=> [[3,1,0],[2,1],[2]]
=> [[3,1,0],[2,0],[1]]
=> 7
[2,1,3,1] => [[1,1],[2,3]]
=> [[2,2,0],[2,1],[2]]
=> [[2,2,0],[2,0],[1]]
=> 7
[2,3,1,1] => [[1,1],[2,3]]
=> [[2,2,0],[2,1],[2]]
=> [[2,2,0],[2,0],[1]]
=> 7
[3,1,1,2] => [[1,1,2],[3]]
=> [[3,1,0],[3,0],[2]]
=> [[3,1,0],[1,1],[1]]
=> 7
[3,1,2,1] => [[1,1],[2],[3]]
=> [[2,1,1],[2,1],[2]]
=> [[2,1,1],[1,1],[1]]
=> 7
[1,1,1,1,4] => [[1,1,1,1,4]]
=> [[5,0,0,0],[4,0,0],[4,0],[4]]
=> [[5,0,0,0],[1,0,0],[1,0],[1]]
=> ? = 8
[1,1,1,1,5] => [[1,1,1,1,5]]
=> [[5,0,0,0,0],[4,0,0,0],[4,0,0],[4,0],[4]]
=> ?
=> ? = 9
[1,1,1,5,1] => [[1,1,1,1],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[4,0],[4]]
=> ?
=> ? = 9
[1,1,5,1,1] => [[1,1,1,1],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[4,0],[4]]
=> ?
=> ? = 9
[1,5,1,1,1] => [[1,1,1,1],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[4,0],[4]]
=> ?
=> ? = 9
[5,1,1,1,1] => [[1,1,1,1],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[4,0],[4]]
=> ?
=> ? = 9
[1,1,1,2,4] => [[1,1,1,2,4]]
=> [[5,0,0,0],[4,0,0],[4,0],[3]]
=> [[5,0,0,0],[2,0,0],[1,0],[1]]
=> ? = 9
[1,1,1,2,5] => [[1,1,1,2,5]]
=> [[5,0,0,0,0],[4,0,0,0],[4,0,0],[4,0],[3]]
=> ?
=> ? = 10
[1,1,1,5,2] => [[1,1,1,2],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[4,0],[3]]
=> ?
=> ? = 10
[1,1,2,1,5] => [[1,1,1,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[1,1,2,5,1] => [[1,1,1,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[1,1,5,1,2] => [[1,1,1,2],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[4,0],[3]]
=> ?
=> ? = 10
[1,1,5,2,1] => [[1,1,1],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[1,2,1,1,5] => [[1,1,1,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[1,2,1,5,1] => [[1,1,1],[2,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[1,2,5,1,1] => [[1,1,1],[2,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[1,5,1,1,2] => [[1,1,1,2],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[4,0],[3]]
=> ?
=> ? = 10
[1,5,1,2,1] => [[1,1,1],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[1,5,2,1,1] => [[1,1,1],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[2,1,1,1,5] => [[1,1,1,5],[2]]
=> [[4,1,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[2,1,1,5,1] => [[1,1,1],[2,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[2,1,5,1,1] => [[1,1,1],[2,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[2,5,1,1,1] => [[1,1,1],[2,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[5,1,1,1,2] => [[1,1,1,2],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[4,0],[3]]
=> ?
=> ? = 10
[5,1,1,2,1] => [[1,1,1],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[5,1,2,1,1] => [[1,1,1],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[5,2,1,1,1] => [[1,1,1],[2],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,1],[3]]
=> ?
=> ? = 10
[1,1,1,3,4] => [[1,1,1,3,4]]
=> [[5,0,0,0],[4,0,0],[3,0],[3]]
=> [[5,0,0,0],[2,0,0],[2,0],[1]]
=> ? = 10
[1,1,1,3,5] => [[1,1,1,3,5]]
=> [[5,0,0,0,0],[4,0,0,0],[4,0,0],[3,0],[3]]
=> ?
=> ? = 11
[1,1,1,5,3] => [[1,1,1,3],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[3,0],[3]]
=> ?
=> ? = 11
[1,1,3,1,5] => [[1,1,1,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[1,1,3,5,1] => [[1,1,1,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[1,1,5,1,3] => [[1,1,1,3],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[3,0],[3]]
=> ?
=> ? = 11
[1,1,5,3,1] => [[1,1,1],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[1,3,1,1,5] => [[1,1,1,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[1,3,1,5,1] => [[1,1,1],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[1,3,5,1,1] => [[1,1,1],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[1,5,1,1,3] => [[1,1,1,3],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[3,0],[3]]
=> ?
=> ? = 11
[1,5,1,3,1] => [[1,1,1],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[1,5,3,1,1] => [[1,1,1],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[3,1,1,1,5] => [[1,1,1,5],[3]]
=> [[4,1,0,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[3,1,1,5,1] => [[1,1,1],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[3,1,5,1,1] => [[1,1,1],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[3,5,1,1,1] => [[1,1,1],[3,5]]
=> [[3,2,0,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[5,1,1,1,3] => [[1,1,1,3],[5]]
=> [[4,1,0,0,0],[4,0,0,0],[4,0,0],[3,0],[3]]
=> ?
=> ? = 11
[5,1,1,3,1] => [[1,1,1],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[5,1,3,1,1] => [[1,1,1],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[5,3,1,1,1] => [[1,1,1],[3],[5]]
=> [[3,1,1,0,0],[3,1,0,0],[3,1,0],[3,0],[3]]
=> ?
=> ? = 11
[1,1,1,4,4] => [[1,1,1,4,4]]
=> [[5,0,0,0],[3,0,0],[3,0],[3]]
=> [[5,0,0,0],[2,0,0],[2,0],[2]]
=> ? = 11
[1,1,1,4,5] => [[1,1,1,4,5]]
=> [[5,0,0,0,0],[4,0,0,0],[3,0,0],[3,0],[3]]
=> ?
=> ? = 12
Description
The sum of the entries of the Gelfand-Tsetlin pattern.
Matching statistic: St000548
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 93%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000548: Integer partitions ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 93%
Values
[1] => [1] => [1]
=> 1
[1,1] => [1,1] => [1,1]
=> 2
[1,2] => [1,2] => [2,1]
=> 3
[2,1] => [2,1] => [2,1]
=> 3
[1,1,1] => [1,1,1] => [1,1,1]
=> 3
[1,1,2] => [1,1,2] => [2,1,1]
=> 4
[1,2,1] => [1,2,1] => [2,1,1]
=> 4
[2,1,1] => [2,1,1] => [2,1,1]
=> 4
[1,1,3] => [1,1,3] => [3,1,1]
=> 5
[1,3,1] => [1,3,1] => [3,1,1]
=> 5
[3,1,1] => [3,1,1] => [3,1,1]
=> 5
[1,2,2] => [1,2,2] => [2,2,1]
=> 5
[2,1,2] => [2,1,2] => [2,2,1]
=> 5
[2,2,1] => [2,2,1] => [2,2,1]
=> 5
[1,2,3] => [1,2,3] => [3,2,1]
=> 6
[1,3,2] => [1,3,2] => [3,2,1]
=> 6
[2,1,3] => [2,1,3] => [3,2,1]
=> 6
[2,3,1] => [2,3,1] => [3,2,1]
=> 6
[3,1,2] => [3,1,2] => [3,2,1]
=> 6
[3,2,1] => [3,2,1] => [3,2,1]
=> 6
[1,1,1,1] => [1,1,1,1] => [1,1,1,1]
=> 4
[1,1,1,2] => [1,1,1,2] => [2,1,1,1]
=> 5
[1,1,2,1] => [1,1,2,1] => [2,1,1,1]
=> 5
[1,2,1,1] => [1,2,1,1] => [2,1,1,1]
=> 5
[2,1,1,1] => [2,1,1,1] => [2,1,1,1]
=> 5
[1,1,1,3] => [1,1,1,3] => [3,1,1,1]
=> 6
[1,1,3,1] => [1,1,3,1] => [3,1,1,1]
=> 6
[1,3,1,1] => [1,3,1,1] => [3,1,1,1]
=> 6
[3,1,1,1] => [3,1,1,1] => [3,1,1,1]
=> 6
[1,1,1,4] => [1,1,1,4] => [4,1,1,1]
=> 7
[1,1,4,1] => [1,1,4,1] => [4,1,1,1]
=> 7
[1,4,1,1] => [1,4,1,1] => [4,1,1,1]
=> 7
[4,1,1,1] => [4,1,1,1] => [4,1,1,1]
=> 7
[1,1,2,2] => [1,1,2,2] => [2,2,1,1]
=> 6
[1,2,1,2] => [1,2,1,2] => [2,2,1,1]
=> 6
[1,2,2,1] => [1,2,2,1] => [2,2,1,1]
=> 6
[2,1,1,2] => [2,1,1,2] => [2,2,1,1]
=> 6
[2,1,2,1] => [2,1,2,1] => [2,2,1,1]
=> 6
[2,2,1,1] => [2,2,1,1] => [2,2,1,1]
=> 6
[1,1,2,3] => [1,1,2,3] => [3,2,1,1]
=> 7
[1,1,3,2] => [1,1,3,2] => [3,2,1,1]
=> 7
[1,2,1,3] => [1,2,1,3] => [3,2,1,1]
=> 7
[1,2,3,1] => [1,2,3,1] => [3,2,1,1]
=> 7
[1,3,1,2] => [1,3,1,2] => [3,2,1,1]
=> 7
[1,3,2,1] => [1,3,2,1] => [3,2,1,1]
=> 7
[2,1,1,3] => [2,1,1,3] => [3,2,1,1]
=> 7
[2,1,3,1] => [2,1,3,1] => [3,2,1,1]
=> 7
[2,3,1,1] => [2,3,1,1] => [3,2,1,1]
=> 7
[3,1,1,2] => [3,1,1,2] => [3,2,1,1]
=> 7
[3,1,2,1] => [3,1,2,1] => [3,2,1,1]
=> 7
[1,1,1,3,5] => [1,1,1,3,5] => ?
=> ? = 11
[1,1,1,5,3] => [1,1,1,5,3] => ?
=> ? = 11
[1,1,3,1,5] => [1,1,3,1,5] => ?
=> ? = 11
[1,1,3,5,1] => [1,1,3,5,1] => ?
=> ? = 11
[1,1,5,1,3] => [1,1,5,1,3] => ?
=> ? = 11
[1,1,5,3,1] => [1,1,5,3,1] => ?
=> ? = 11
[1,3,1,1,5] => [1,3,1,1,5] => ?
=> ? = 11
[1,3,5,1,1] => [1,3,5,1,1] => ?
=> ? = 11
[1,5,1,1,3] => [1,5,1,1,3] => ?
=> ? = 11
[1,5,3,1,1] => [1,5,3,1,1] => ?
=> ? = 11
[3,1,1,1,5] => [3,1,1,1,5] => ?
=> ? = 11
[3,1,1,5,1] => [3,1,1,5,1] => ?
=> ? = 11
[3,1,5,1,1] => [3,1,5,1,1] => ?
=> ? = 11
[3,5,1,1,1] => [3,5,1,1,1] => ?
=> ? = 11
[5,1,1,1,3] => [5,1,1,1,3] => ?
=> ? = 11
[5,1,1,3,1] => [5,1,1,3,1] => ?
=> ? = 11
[5,1,3,1,1] => [5,1,3,1,1] => ?
=> ? = 11
[5,3,1,1,1] => [5,3,1,1,1] => ?
=> ? = 11
[1,1,1,4,4] => [1,1,1,4,4] => ?
=> ? = 11
[1,1,4,1,4] => [1,1,4,1,4] => ?
=> ? = 11
[1,1,4,4,1] => [1,1,4,4,1] => ?
=> ? = 11
[1,4,1,1,4] => [1,4,1,1,4] => ?
=> ? = 11
[1,4,4,1,1] => [1,4,4,1,1] => ?
=> ? = 11
[4,1,1,1,4] => [4,1,1,1,4] => ?
=> ? = 11
[4,1,1,4,1] => [4,1,1,4,1] => ?
=> ? = 11
[4,1,4,1,1] => [4,1,4,1,1] => ?
=> ? = 11
[4,4,1,1,1] => [4,4,1,1,1] => ?
=> ? = 11
[1,1,1,4,5] => [1,1,1,4,5] => ?
=> ? = 12
[1,1,1,5,4] => [1,1,1,5,4] => ?
=> ? = 12
[1,1,5,1,4] => [1,1,5,1,4] => ?
=> ? = 12
[1,1,5,4,1] => [1,1,5,4,1] => ?
=> ? = 12
[1,4,1,1,5] => [1,4,1,1,5] => ?
=> ? = 12
[1,4,1,5,1] => [1,4,1,5,1] => ?
=> ? = 12
[1,4,5,1,1] => [1,4,5,1,1] => ?
=> ? = 12
[1,5,1,1,4] => [1,5,1,1,4] => ?
=> ? = 12
[1,5,1,4,1] => [1,5,1,4,1] => ?
=> ? = 12
[4,1,1,1,5] => [4,1,1,1,5] => ?
=> ? = 12
[4,1,1,5,1] => [4,1,1,5,1] => ?
=> ? = 12
[4,1,5,1,1] => [4,1,5,1,1] => ?
=> ? = 12
[4,5,1,1,1] => [4,5,1,1,1] => ?
=> ? = 12
[5,1,1,1,4] => [5,1,1,1,4] => ?
=> ? = 12
[5,1,1,4,1] => [5,1,1,4,1] => ?
=> ? = 12
[5,4,1,1,1] => [5,4,1,1,1] => ?
=> ? = 12
[1,1,2,2,5] => [1,1,2,2,5] => ?
=> ? = 11
[1,1,2,5,2] => [1,1,2,5,2] => ?
=> ? = 11
[1,1,5,2,2] => [1,1,5,2,2] => ?
=> ? = 11
[1,2,1,2,5] => [1,2,1,2,5] => ?
=> ? = 11
[1,2,1,5,2] => [1,2,1,5,2] => ?
=> ? = 11
[1,2,2,1,5] => [1,2,2,1,5] => ?
=> ? = 11
[1,2,2,5,1] => [1,2,2,5,1] => ?
=> ? = 11
Description
The number of different non-empty partial sums of an integer partition.
Matching statistic: St000293
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 93%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 93%
Values
[1] => [1] => [1]
=> 10 => 1
[1,1] => [1,1] => [1,1]
=> 110 => 2
[1,2] => [1,2] => [2,1]
=> 1010 => 3
[2,1] => [2,1] => [2,1]
=> 1010 => 3
[1,1,1] => [1,1,1] => [1,1,1]
=> 1110 => 3
[1,1,2] => [1,1,2] => [2,1,1]
=> 10110 => 4
[1,2,1] => [1,2,1] => [2,1,1]
=> 10110 => 4
[2,1,1] => [2,1,1] => [2,1,1]
=> 10110 => 4
[1,1,3] => [1,1,3] => [3,1,1]
=> 100110 => 5
[1,3,1] => [1,3,1] => [3,1,1]
=> 100110 => 5
[3,1,1] => [3,1,1] => [3,1,1]
=> 100110 => 5
[1,2,2] => [1,2,2] => [2,2,1]
=> 11010 => 5
[2,1,2] => [2,1,2] => [2,2,1]
=> 11010 => 5
[2,2,1] => [2,2,1] => [2,2,1]
=> 11010 => 5
[1,2,3] => [1,2,3] => [3,2,1]
=> 101010 => 6
[1,3,2] => [1,3,2] => [3,2,1]
=> 101010 => 6
[2,1,3] => [2,1,3] => [3,2,1]
=> 101010 => 6
[2,3,1] => [2,3,1] => [3,2,1]
=> 101010 => 6
[3,1,2] => [3,1,2] => [3,2,1]
=> 101010 => 6
[3,2,1] => [3,2,1] => [3,2,1]
=> 101010 => 6
[1,1,1,1] => [1,1,1,1] => [1,1,1,1]
=> 11110 => 4
[1,1,1,2] => [1,1,1,2] => [2,1,1,1]
=> 101110 => 5
[1,1,2,1] => [1,1,2,1] => [2,1,1,1]
=> 101110 => 5
[1,2,1,1] => [1,2,1,1] => [2,1,1,1]
=> 101110 => 5
[2,1,1,1] => [2,1,1,1] => [2,1,1,1]
=> 101110 => 5
[1,1,1,3] => [1,1,1,3] => [3,1,1,1]
=> 1001110 => 6
[1,1,3,1] => [1,1,3,1] => [3,1,1,1]
=> 1001110 => 6
[1,3,1,1] => [1,3,1,1] => [3,1,1,1]
=> 1001110 => 6
[3,1,1,1] => [3,1,1,1] => [3,1,1,1]
=> 1001110 => 6
[1,1,1,4] => [1,1,1,4] => [4,1,1,1]
=> 10001110 => 7
[1,1,4,1] => [1,1,4,1] => [4,1,1,1]
=> 10001110 => 7
[1,4,1,1] => [1,4,1,1] => [4,1,1,1]
=> 10001110 => 7
[4,1,1,1] => [4,1,1,1] => [4,1,1,1]
=> 10001110 => 7
[1,1,2,2] => [1,1,2,2] => [2,2,1,1]
=> 110110 => 6
[1,2,1,2] => [1,2,1,2] => [2,2,1,1]
=> 110110 => 6
[1,2,2,1] => [1,2,2,1] => [2,2,1,1]
=> 110110 => 6
[2,1,1,2] => [2,1,1,2] => [2,2,1,1]
=> 110110 => 6
[2,1,2,1] => [2,1,2,1] => [2,2,1,1]
=> 110110 => 6
[2,2,1,1] => [2,2,1,1] => [2,2,1,1]
=> 110110 => 6
[1,1,2,3] => [1,1,2,3] => [3,2,1,1]
=> 1010110 => 7
[1,1,3,2] => [1,1,3,2] => [3,2,1,1]
=> 1010110 => 7
[1,2,1,3] => [1,2,1,3] => [3,2,1,1]
=> 1010110 => 7
[1,2,3,1] => [1,2,3,1] => [3,2,1,1]
=> 1010110 => 7
[1,3,1,2] => [1,3,1,2] => [3,2,1,1]
=> 1010110 => 7
[1,3,2,1] => [1,3,2,1] => [3,2,1,1]
=> 1010110 => 7
[2,1,1,3] => [2,1,1,3] => [3,2,1,1]
=> 1010110 => 7
[2,1,3,1] => [2,1,3,1] => [3,2,1,1]
=> 1010110 => 7
[2,3,1,1] => [2,3,1,1] => [3,2,1,1]
=> 1010110 => 7
[3,1,1,2] => [3,1,1,2] => [3,2,1,1]
=> 1010110 => 7
[3,1,2,1] => [3,1,2,1] => [3,2,1,1]
=> 1010110 => 7
[1,1,1,3,5] => [1,1,1,3,5] => ?
=> ? => ? = 11
[1,1,1,5,3] => [1,1,1,5,3] => ?
=> ? => ? = 11
[1,1,3,1,5] => [1,1,3,1,5] => ?
=> ? => ? = 11
[1,1,3,5,1] => [1,1,3,5,1] => ?
=> ? => ? = 11
[1,1,5,1,3] => [1,1,5,1,3] => ?
=> ? => ? = 11
[1,1,5,3,1] => [1,1,5,3,1] => ?
=> ? => ? = 11
[1,3,1,1,5] => [1,3,1,1,5] => ?
=> ? => ? = 11
[1,3,5,1,1] => [1,3,5,1,1] => ?
=> ? => ? = 11
[1,5,1,1,3] => [1,5,1,1,3] => ?
=> ? => ? = 11
[1,5,3,1,1] => [1,5,3,1,1] => ?
=> ? => ? = 11
[3,1,1,1,5] => [3,1,1,1,5] => ?
=> ? => ? = 11
[3,1,1,5,1] => [3,1,1,5,1] => ?
=> ? => ? = 11
[3,1,5,1,1] => [3,1,5,1,1] => ?
=> ? => ? = 11
[3,5,1,1,1] => [3,5,1,1,1] => ?
=> ? => ? = 11
[5,1,1,1,3] => [5,1,1,1,3] => ?
=> ? => ? = 11
[5,1,1,3,1] => [5,1,1,3,1] => ?
=> ? => ? = 11
[5,1,3,1,1] => [5,1,3,1,1] => ?
=> ? => ? = 11
[5,3,1,1,1] => [5,3,1,1,1] => ?
=> ? => ? = 11
[1,1,1,4,4] => [1,1,1,4,4] => ?
=> ? => ? = 11
[1,1,4,1,4] => [1,1,4,1,4] => ?
=> ? => ? = 11
[1,1,4,4,1] => [1,1,4,4,1] => ?
=> ? => ? = 11
[1,4,1,1,4] => [1,4,1,1,4] => ?
=> ? => ? = 11
[1,4,4,1,1] => [1,4,4,1,1] => ?
=> ? => ? = 11
[4,1,1,1,4] => [4,1,1,1,4] => ?
=> ? => ? = 11
[4,1,1,4,1] => [4,1,1,4,1] => ?
=> ? => ? = 11
[4,1,4,1,1] => [4,1,4,1,1] => ?
=> ? => ? = 11
[4,4,1,1,1] => [4,4,1,1,1] => ?
=> ? => ? = 11
[1,1,1,4,5] => [1,1,1,4,5] => ?
=> ? => ? = 12
[1,1,1,5,4] => [1,1,1,5,4] => ?
=> ? => ? = 12
[1,1,5,1,4] => [1,1,5,1,4] => ?
=> ? => ? = 12
[1,1,5,4,1] => [1,1,5,4,1] => ?
=> ? => ? = 12
[1,4,1,1,5] => [1,4,1,1,5] => ?
=> ? => ? = 12
[1,4,1,5,1] => [1,4,1,5,1] => ?
=> ? => ? = 12
[1,4,5,1,1] => [1,4,5,1,1] => ?
=> ? => ? = 12
[1,5,1,1,4] => [1,5,1,1,4] => ?
=> ? => ? = 12
[1,5,1,4,1] => [1,5,1,4,1] => ?
=> ? => ? = 12
[4,1,1,1,5] => [4,1,1,1,5] => ?
=> ? => ? = 12
[4,1,1,5,1] => [4,1,1,5,1] => ?
=> ? => ? = 12
[4,1,5,1,1] => [4,1,5,1,1] => ?
=> ? => ? = 12
[4,5,1,1,1] => [4,5,1,1,1] => ?
=> ? => ? = 12
[5,1,1,1,4] => [5,1,1,1,4] => ?
=> ? => ? = 12
[5,1,1,4,1] => [5,1,1,4,1] => ?
=> ? => ? = 12
[5,4,1,1,1] => [5,4,1,1,1] => ?
=> ? => ? = 12
[1,1,2,2,5] => [1,1,2,2,5] => ?
=> ? => ? = 11
[1,1,2,5,2] => [1,1,2,5,2] => ?
=> ? => ? = 11
[1,1,5,2,2] => [1,1,5,2,2] => ?
=> ? => ? = 11
[1,2,1,2,5] => [1,2,1,2,5] => ?
=> ? => ? = 11
[1,2,1,5,2] => [1,2,1,5,2] => ?
=> ? => ? = 11
[1,2,2,1,5] => [1,2,2,1,5] => ?
=> ? => ? = 11
[1,2,2,5,1] => [1,2,2,5,1] => ?
=> ? => ? = 11
Description
The number of inversions of a binary word.
Matching statistic: St000738
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 93%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 93%
Values
[1] => [1] => [1]
=> [[1]]
=> 1
[1,1] => [1,1] => [1,1]
=> [[1],[2]]
=> 2
[1,2] => [1,2] => [2,1]
=> [[1,2],[3]]
=> 3
[2,1] => [2,1] => [2,1]
=> [[1,2],[3]]
=> 3
[1,1,1] => [1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,1,2] => [1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 4
[1,2,1] => [1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 4
[2,1,1] => [2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 4
[1,1,3] => [1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 5
[1,3,1] => [1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 5
[3,1,1] => [3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 5
[1,2,2] => [1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 5
[2,1,2] => [2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 5
[2,2,1] => [2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 5
[1,2,3] => [1,2,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 6
[1,3,2] => [1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 6
[2,1,3] => [2,1,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 6
[2,3,1] => [2,3,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 6
[3,1,2] => [3,1,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 6
[3,2,1] => [3,2,1] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 6
[1,1,1,1] => [1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[1,1,1,2] => [1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 5
[1,1,2,1] => [1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 5
[1,2,1,1] => [1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 5
[2,1,1,1] => [2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 5
[1,1,1,3] => [1,1,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 6
[1,1,3,1] => [1,1,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 6
[1,3,1,1] => [1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 6
[3,1,1,1] => [3,1,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 6
[1,1,1,4] => [1,1,1,4] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 7
[1,1,4,1] => [1,1,4,1] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 7
[1,4,1,1] => [1,4,1,1] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 7
[4,1,1,1] => [4,1,1,1] => [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 7
[1,1,2,2] => [1,1,2,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 6
[1,2,1,2] => [1,2,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 6
[1,2,2,1] => [1,2,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 6
[2,1,1,2] => [2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 6
[2,1,2,1] => [2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 6
[2,2,1,1] => [2,2,1,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 6
[1,1,2,3] => [1,1,2,3] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[1,1,3,2] => [1,1,3,2] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[1,2,1,3] => [1,2,1,3] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[1,2,3,1] => [1,2,3,1] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[1,3,1,2] => [1,3,1,2] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[1,3,2,1] => [1,3,2,1] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[2,1,1,3] => [2,1,1,3] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[2,1,3,1] => [2,1,3,1] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[2,3,1,1] => [2,3,1,1] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[3,1,1,2] => [3,1,1,2] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[3,1,2,1] => [3,1,2,1] => [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 7
[1,1,1,3,5] => [1,1,1,3,5] => ?
=> ?
=> ? = 11
[1,1,1,5,3] => [1,1,1,5,3] => ?
=> ?
=> ? = 11
[1,1,3,1,5] => [1,1,3,1,5] => ?
=> ?
=> ? = 11
[1,1,3,5,1] => [1,1,3,5,1] => ?
=> ?
=> ? = 11
[1,1,5,1,3] => [1,1,5,1,3] => ?
=> ?
=> ? = 11
[1,1,5,3,1] => [1,1,5,3,1] => ?
=> ?
=> ? = 11
[1,3,1,1,5] => [1,3,1,1,5] => ?
=> ?
=> ? = 11
[1,3,5,1,1] => [1,3,5,1,1] => ?
=> ?
=> ? = 11
[1,5,1,1,3] => [1,5,1,1,3] => ?
=> ?
=> ? = 11
[1,5,3,1,1] => [1,5,3,1,1] => ?
=> ?
=> ? = 11
[3,1,1,1,5] => [3,1,1,1,5] => ?
=> ?
=> ? = 11
[3,1,1,5,1] => [3,1,1,5,1] => ?
=> ?
=> ? = 11
[3,1,5,1,1] => [3,1,5,1,1] => ?
=> ?
=> ? = 11
[3,5,1,1,1] => [3,5,1,1,1] => ?
=> ?
=> ? = 11
[5,1,1,1,3] => [5,1,1,1,3] => ?
=> ?
=> ? = 11
[5,1,1,3,1] => [5,1,1,3,1] => ?
=> ?
=> ? = 11
[5,1,3,1,1] => [5,1,3,1,1] => ?
=> ?
=> ? = 11
[5,3,1,1,1] => [5,3,1,1,1] => ?
=> ?
=> ? = 11
[1,1,1,4,4] => [1,1,1,4,4] => ?
=> ?
=> ? = 11
[1,1,4,1,4] => [1,1,4,1,4] => ?
=> ?
=> ? = 11
[1,1,4,4,1] => [1,1,4,4,1] => ?
=> ?
=> ? = 11
[1,4,1,1,4] => [1,4,1,1,4] => ?
=> ?
=> ? = 11
[1,4,4,1,1] => [1,4,4,1,1] => ?
=> ?
=> ? = 11
[4,1,1,1,4] => [4,1,1,1,4] => ?
=> ?
=> ? = 11
[4,1,1,4,1] => [4,1,1,4,1] => ?
=> ?
=> ? = 11
[4,1,4,1,1] => [4,1,4,1,1] => ?
=> ?
=> ? = 11
[4,4,1,1,1] => [4,4,1,1,1] => ?
=> ?
=> ? = 11
[1,1,1,4,5] => [1,1,1,4,5] => ?
=> ?
=> ? = 12
[1,1,1,5,4] => [1,1,1,5,4] => ?
=> ?
=> ? = 12
[1,1,5,1,4] => [1,1,5,1,4] => ?
=> ?
=> ? = 12
[1,1,5,4,1] => [1,1,5,4,1] => ?
=> ?
=> ? = 12
[1,4,1,1,5] => [1,4,1,1,5] => ?
=> ?
=> ? = 12
[1,4,1,5,1] => [1,4,1,5,1] => ?
=> ?
=> ? = 12
[1,4,5,1,1] => [1,4,5,1,1] => ?
=> ?
=> ? = 12
[1,5,1,1,4] => [1,5,1,1,4] => ?
=> ?
=> ? = 12
[1,5,1,4,1] => [1,5,1,4,1] => ?
=> ?
=> ? = 12
[4,1,1,1,5] => [4,1,1,1,5] => ?
=> ?
=> ? = 12
[4,1,1,5,1] => [4,1,1,5,1] => ?
=> ?
=> ? = 12
[4,1,5,1,1] => [4,1,5,1,1] => ?
=> ?
=> ? = 12
[4,5,1,1,1] => [4,5,1,1,1] => ?
=> ?
=> ? = 12
[5,1,1,1,4] => [5,1,1,1,4] => ?
=> ?
=> ? = 12
[5,1,1,4,1] => [5,1,1,4,1] => ?
=> ?
=> ? = 12
[5,4,1,1,1] => [5,4,1,1,1] => ?
=> ?
=> ? = 12
[1,1,2,2,5] => [1,1,2,2,5] => ?
=> ?
=> ? = 11
[1,1,2,5,2] => [1,1,2,5,2] => ?
=> ?
=> ? = 11
[1,1,5,2,2] => [1,1,5,2,2] => ?
=> ?
=> ? = 11
[1,2,1,2,5] => [1,2,1,2,5] => ?
=> ?
=> ? = 11
[1,2,1,5,2] => [1,2,1,5,2] => ?
=> ?
=> ? = 11
[1,2,2,1,5] => [1,2,2,1,5] => ?
=> ?
=> ? = 11
[1,2,2,5,1] => [1,2,2,5,1] => ?
=> ?
=> ? = 11
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St001034
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 93%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 93%
Values
[1] => [1] => [1]
=> [1,0]
=> 1
[1,1] => [1,1] => [1,1]
=> [1,1,0,0]
=> 2
[1,2] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 3
[2,1] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 3
[1,1,1] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 3
[1,1,2] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,2,1] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[2,1,1] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 4
[1,1,3] => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[1,3,1] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[3,1,1] => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 5
[1,2,2] => [1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5
[2,1,2] => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5
[2,2,1] => [2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 5
[1,2,3] => [1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[1,3,2] => [1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[2,1,3] => [2,1,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[2,3,1] => [2,3,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[3,1,2] => [3,1,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[3,2,1] => [3,2,1] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
[1,1,1,1] => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 4
[1,1,1,2] => [1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 5
[1,1,2,1] => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 5
[1,2,1,1] => [1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 5
[2,1,1,1] => [2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 5
[1,1,1,3] => [1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 6
[1,1,3,1] => [1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 6
[1,3,1,1] => [1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 6
[3,1,1,1] => [3,1,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 6
[1,1,1,4] => [1,1,1,4] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 7
[1,1,4,1] => [1,1,4,1] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 7
[1,4,1,1] => [1,4,1,1] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 7
[4,1,1,1] => [4,1,1,1] => [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 7
[1,1,2,2] => [1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 6
[1,2,1,2] => [1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 6
[1,2,2,1] => [1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 6
[2,1,1,2] => [2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 6
[2,1,2,1] => [2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 6
[2,2,1,1] => [2,2,1,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 6
[1,1,2,3] => [1,1,2,3] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[1,1,3,2] => [1,1,3,2] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[1,2,1,3] => [1,2,1,3] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[1,2,3,1] => [1,2,3,1] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[1,3,1,2] => [1,3,1,2] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[1,3,2,1] => [1,3,2,1] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[2,1,1,3] => [2,1,1,3] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[2,1,3,1] => [2,1,3,1] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[2,3,1,1] => [2,3,1,1] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[3,1,1,2] => [3,1,1,2] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[3,1,2,1] => [3,1,2,1] => [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 7
[1,1,1,3,5] => [1,1,1,3,5] => ?
=> ?
=> ? = 11
[1,1,1,5,3] => [1,1,1,5,3] => ?
=> ?
=> ? = 11
[1,1,3,1,5] => [1,1,3,1,5] => ?
=> ?
=> ? = 11
[1,1,3,5,1] => [1,1,3,5,1] => ?
=> ?
=> ? = 11
[1,1,5,1,3] => [1,1,5,1,3] => ?
=> ?
=> ? = 11
[1,1,5,3,1] => [1,1,5,3,1] => ?
=> ?
=> ? = 11
[1,3,1,1,5] => [1,3,1,1,5] => ?
=> ?
=> ? = 11
[1,3,5,1,1] => [1,3,5,1,1] => ?
=> ?
=> ? = 11
[1,5,1,1,3] => [1,5,1,1,3] => ?
=> ?
=> ? = 11
[1,5,3,1,1] => [1,5,3,1,1] => ?
=> ?
=> ? = 11
[3,1,1,1,5] => [3,1,1,1,5] => ?
=> ?
=> ? = 11
[3,1,1,5,1] => [3,1,1,5,1] => ?
=> ?
=> ? = 11
[3,1,5,1,1] => [3,1,5,1,1] => ?
=> ?
=> ? = 11
[3,5,1,1,1] => [3,5,1,1,1] => ?
=> ?
=> ? = 11
[5,1,1,1,3] => [5,1,1,1,3] => ?
=> ?
=> ? = 11
[5,1,1,3,1] => [5,1,1,3,1] => ?
=> ?
=> ? = 11
[5,1,3,1,1] => [5,1,3,1,1] => ?
=> ?
=> ? = 11
[5,3,1,1,1] => [5,3,1,1,1] => ?
=> ?
=> ? = 11
[1,1,1,4,4] => [1,1,1,4,4] => ?
=> ?
=> ? = 11
[1,1,4,1,4] => [1,1,4,1,4] => ?
=> ?
=> ? = 11
[1,1,4,4,1] => [1,1,4,4,1] => ?
=> ?
=> ? = 11
[1,4,1,1,4] => [1,4,1,1,4] => ?
=> ?
=> ? = 11
[1,4,4,1,1] => [1,4,4,1,1] => ?
=> ?
=> ? = 11
[4,1,1,1,4] => [4,1,1,1,4] => ?
=> ?
=> ? = 11
[4,1,1,4,1] => [4,1,1,4,1] => ?
=> ?
=> ? = 11
[4,1,4,1,1] => [4,1,4,1,1] => ?
=> ?
=> ? = 11
[4,4,1,1,1] => [4,4,1,1,1] => ?
=> ?
=> ? = 11
[1,1,1,4,5] => [1,1,1,4,5] => ?
=> ?
=> ? = 12
[1,1,1,5,4] => [1,1,1,5,4] => ?
=> ?
=> ? = 12
[1,1,5,1,4] => [1,1,5,1,4] => ?
=> ?
=> ? = 12
[1,1,5,4,1] => [1,1,5,4,1] => ?
=> ?
=> ? = 12
[1,4,1,1,5] => [1,4,1,1,5] => ?
=> ?
=> ? = 12
[1,4,1,5,1] => [1,4,1,5,1] => ?
=> ?
=> ? = 12
[1,4,5,1,1] => [1,4,5,1,1] => ?
=> ?
=> ? = 12
[1,5,1,1,4] => [1,5,1,1,4] => ?
=> ?
=> ? = 12
[1,5,1,4,1] => [1,5,1,4,1] => ?
=> ?
=> ? = 12
[4,1,1,1,5] => [4,1,1,1,5] => ?
=> ?
=> ? = 12
[4,1,1,5,1] => [4,1,1,5,1] => ?
=> ?
=> ? = 12
[4,1,5,1,1] => [4,1,5,1,1] => ?
=> ?
=> ? = 12
[4,5,1,1,1] => [4,5,1,1,1] => ?
=> ?
=> ? = 12
[5,1,1,1,4] => [5,1,1,1,4] => ?
=> ?
=> ? = 12
[5,1,1,4,1] => [5,1,1,4,1] => ?
=> ?
=> ? = 12
[5,4,1,1,1] => [5,4,1,1,1] => ?
=> ?
=> ? = 12
[1,1,2,2,5] => [1,1,2,2,5] => ?
=> ?
=> ? = 11
[1,1,2,5,2] => [1,1,2,5,2] => ?
=> ?
=> ? = 11
[1,1,5,2,2] => [1,1,5,2,2] => ?
=> ?
=> ? = 11
[1,2,1,2,5] => [1,2,1,2,5] => ?
=> ?
=> ? = 11
[1,2,1,5,2] => [1,2,1,5,2] => ?
=> ?
=> ? = 11
[1,2,2,1,5] => [1,2,2,1,5] => ?
=> ?
=> ? = 11
[1,2,2,5,1] => [1,2,2,5,1] => ?
=> ?
=> ? = 11
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000228
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 71%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1]
=> 1
[1,1] => [1,1] => [1,1]
=> 2
[1,2] => [1,2] => [2,1]
=> 3
[2,1] => [2,1] => [2,1]
=> 3
[1,1,1] => [1,1,1] => [1,1,1]
=> 3
[1,1,2] => [1,1,2] => [2,1,1]
=> 4
[1,2,1] => [1,2,1] => [2,1,1]
=> 4
[2,1,1] => [2,1,1] => [2,1,1]
=> 4
[1,1,3] => [1,1,3] => [3,1,1]
=> 5
[1,3,1] => [1,3,1] => [3,1,1]
=> 5
[3,1,1] => [3,1,1] => [3,1,1]
=> 5
[1,2,2] => [1,2,2] => [2,2,1]
=> 5
[2,1,2] => [2,1,2] => [2,2,1]
=> 5
[2,2,1] => [2,2,1] => [2,2,1]
=> 5
[1,2,3] => [1,2,3] => [3,2,1]
=> 6
[1,3,2] => [1,3,2] => [3,2,1]
=> 6
[2,1,3] => [2,1,3] => [3,2,1]
=> 6
[2,3,1] => [2,3,1] => [3,2,1]
=> 6
[3,1,2] => [3,1,2] => [3,2,1]
=> 6
[3,2,1] => [3,2,1] => [3,2,1]
=> 6
[1,1,1,1] => [1,1,1,1] => [1,1,1,1]
=> 4
[1,1,1,2] => [1,1,1,2] => [2,1,1,1]
=> 5
[1,1,2,1] => [1,1,2,1] => [2,1,1,1]
=> 5
[1,2,1,1] => [1,2,1,1] => [2,1,1,1]
=> 5
[2,1,1,1] => [2,1,1,1] => [2,1,1,1]
=> 5
[1,1,1,3] => [1,1,1,3] => [3,1,1,1]
=> 6
[1,1,3,1] => [1,1,3,1] => [3,1,1,1]
=> 6
[1,3,1,1] => [1,3,1,1] => [3,1,1,1]
=> 6
[3,1,1,1] => [3,1,1,1] => [3,1,1,1]
=> 6
[1,1,1,4] => [1,1,1,4] => [4,1,1,1]
=> 7
[1,1,4,1] => [1,1,4,1] => [4,1,1,1]
=> 7
[1,4,1,1] => [1,4,1,1] => [4,1,1,1]
=> 7
[4,1,1,1] => [4,1,1,1] => [4,1,1,1]
=> 7
[1,1,2,2] => [1,1,2,2] => [2,2,1,1]
=> 6
[1,2,1,2] => [1,2,1,2] => [2,2,1,1]
=> 6
[1,2,2,1] => [1,2,2,1] => [2,2,1,1]
=> 6
[2,1,1,2] => [2,1,1,2] => [2,2,1,1]
=> 6
[2,1,2,1] => [2,1,2,1] => [2,2,1,1]
=> 6
[2,2,1,1] => [2,2,1,1] => [2,2,1,1]
=> 6
[1,1,2,3] => [1,1,2,3] => [3,2,1,1]
=> 7
[1,1,3,2] => [1,1,3,2] => [3,2,1,1]
=> 7
[1,2,1,3] => [1,2,1,3] => [3,2,1,1]
=> 7
[1,2,3,1] => [1,2,3,1] => [3,2,1,1]
=> 7
[1,3,1,2] => [1,3,1,2] => [3,2,1,1]
=> 7
[1,3,2,1] => [1,3,2,1] => [3,2,1,1]
=> 7
[2,1,1,3] => [2,1,1,3] => [3,2,1,1]
=> 7
[2,1,3,1] => [2,1,3,1] => [3,2,1,1]
=> 7
[2,3,1,1] => [2,3,1,1] => [3,2,1,1]
=> 7
[3,1,1,2] => [3,1,1,2] => [3,2,1,1]
=> 7
[3,1,2,1] => [3,1,2,1] => [3,2,1,1]
=> 7
[1,1,1,3,5] => [1,1,1,3,5] => ?
=> ? = 11
[1,1,1,5,3] => [1,1,1,5,3] => ?
=> ? = 11
[1,1,3,1,5] => [1,1,3,1,5] => ?
=> ? = 11
[1,1,3,5,1] => [1,1,3,5,1] => ?
=> ? = 11
[1,1,5,1,3] => [1,1,5,1,3] => ?
=> ? = 11
[1,1,5,3,1] => [1,1,5,3,1] => ?
=> ? = 11
[1,3,1,1,5] => [1,3,1,1,5] => ?
=> ? = 11
[1,3,1,5,1] => [1,3,1,5,1] => [5,3,1,1,1]
=> ? = 11
[1,3,5,1,1] => [1,3,5,1,1] => ?
=> ? = 11
[1,5,1,1,3] => [1,5,1,1,3] => ?
=> ? = 11
[1,5,1,3,1] => [1,5,1,3,1] => [5,3,1,1,1]
=> ? = 11
[1,5,3,1,1] => [1,5,3,1,1] => ?
=> ? = 11
[3,1,1,1,5] => [3,1,1,1,5] => ?
=> ? = 11
[3,1,1,5,1] => [3,1,1,5,1] => ?
=> ? = 11
[3,1,5,1,1] => [3,1,5,1,1] => ?
=> ? = 11
[3,5,1,1,1] => [3,5,1,1,1] => ?
=> ? = 11
[5,1,1,1,3] => [5,1,1,1,3] => ?
=> ? = 11
[5,1,1,3,1] => [5,1,1,3,1] => ?
=> ? = 11
[5,1,3,1,1] => [5,1,3,1,1] => ?
=> ? = 11
[5,3,1,1,1] => [5,3,1,1,1] => ?
=> ? = 11
[1,1,1,4,4] => [1,1,1,4,4] => ?
=> ? = 11
[1,1,4,1,4] => [1,1,4,1,4] => ?
=> ? = 11
[1,1,4,4,1] => [1,1,4,4,1] => ?
=> ? = 11
[1,4,1,1,4] => [1,4,1,1,4] => ?
=> ? = 11
[1,4,1,4,1] => [1,4,1,4,1] => [4,4,1,1,1]
=> ? = 11
[1,4,4,1,1] => [1,4,4,1,1] => ?
=> ? = 11
[4,1,1,1,4] => [4,1,1,1,4] => ?
=> ? = 11
[4,1,1,4,1] => [4,1,1,4,1] => ?
=> ? = 11
[4,1,4,1,1] => [4,1,4,1,1] => ?
=> ? = 11
[4,4,1,1,1] => [4,4,1,1,1] => ?
=> ? = 11
[1,1,1,4,5] => [1,1,1,4,5] => ?
=> ? = 12
[1,1,1,5,4] => [1,1,1,5,4] => ?
=> ? = 12
[1,1,4,1,5] => [1,1,4,1,5] => [5,4,1,1,1]
=> ? = 12
[1,1,4,5,1] => [1,1,4,5,1] => [5,4,1,1,1]
=> ? = 12
[1,1,5,1,4] => [1,1,5,1,4] => ?
=> ? = 12
[1,1,5,4,1] => [1,1,5,4,1] => ?
=> ? = 12
[1,4,1,1,5] => [1,4,1,1,5] => ?
=> ? = 12
[1,4,1,5,1] => [1,4,1,5,1] => ?
=> ? = 12
[1,4,5,1,1] => [1,4,5,1,1] => ?
=> ? = 12
[1,5,1,1,4] => [1,5,1,1,4] => ?
=> ? = 12
[1,5,1,4,1] => [1,5,1,4,1] => ?
=> ? = 12
[1,5,4,1,1] => [1,5,4,1,1] => [5,4,1,1,1]
=> ? = 12
[4,1,1,1,5] => [4,1,1,1,5] => ?
=> ? = 12
[4,1,1,5,1] => [4,1,1,5,1] => ?
=> ? = 12
[4,1,5,1,1] => [4,1,5,1,1] => ?
=> ? = 12
[4,5,1,1,1] => [4,5,1,1,1] => ?
=> ? = 12
[5,1,1,1,4] => [5,1,1,1,4] => ?
=> ? = 12
[5,1,1,4,1] => [5,1,1,4,1] => ?
=> ? = 12
[5,1,4,1,1] => [5,1,4,1,1] => [5,4,1,1,1]
=> ? = 12
[5,4,1,1,1] => [5,4,1,1,1] => ?
=> ? = 12
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St001437
Mp00052: Parking functions —to non-decreasing parking function⟶ Parking functions
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 71%
Mp00319: Parking functions —to composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St001437: Binary words ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 71%
Values
[1] => [1] => [1] => 1 => 1
[1,1] => [1,1] => [1,1] => 11 => 2
[1,2] => [1,2] => [1,2] => 110 => 3
[2,1] => [1,2] => [1,2] => 110 => 3
[1,1,1] => [1,1,1] => [1,1,1] => 111 => 3
[1,1,2] => [1,1,2] => [1,1,2] => 1110 => 4
[1,2,1] => [1,1,2] => [1,1,2] => 1110 => 4
[2,1,1] => [1,1,2] => [1,1,2] => 1110 => 4
[1,1,3] => [1,1,3] => [1,1,3] => 11100 => 5
[1,3,1] => [1,1,3] => [1,1,3] => 11100 => 5
[3,1,1] => [1,1,3] => [1,1,3] => 11100 => 5
[1,2,2] => [1,2,2] => [1,2,2] => 11010 => 5
[2,1,2] => [1,2,2] => [1,2,2] => 11010 => 5
[2,2,1] => [1,2,2] => [1,2,2] => 11010 => 5
[1,2,3] => [1,2,3] => [1,2,3] => 110100 => 6
[1,3,2] => [1,2,3] => [1,2,3] => 110100 => 6
[2,1,3] => [1,2,3] => [1,2,3] => 110100 => 6
[2,3,1] => [1,2,3] => [1,2,3] => 110100 => 6
[3,1,2] => [1,2,3] => [1,2,3] => 110100 => 6
[3,2,1] => [1,2,3] => [1,2,3] => 110100 => 6
[1,1,1,1] => [1,1,1,1] => [1,1,1,1] => 1111 => 4
[1,1,1,2] => [1,1,1,2] => [1,1,1,2] => 11110 => 5
[1,1,2,1] => [1,1,1,2] => [1,1,1,2] => 11110 => 5
[1,2,1,1] => [1,1,1,2] => [1,1,1,2] => 11110 => 5
[2,1,1,1] => [1,1,1,2] => [1,1,1,2] => 11110 => 5
[1,1,1,3] => [1,1,1,3] => [1,1,1,3] => 111100 => 6
[1,1,3,1] => [1,1,1,3] => [1,1,1,3] => 111100 => 6
[1,3,1,1] => [1,1,1,3] => [1,1,1,3] => 111100 => 6
[3,1,1,1] => [1,1,1,3] => [1,1,1,3] => 111100 => 6
[1,1,1,4] => [1,1,1,4] => [1,1,1,4] => 1111000 => 7
[1,1,4,1] => [1,1,1,4] => [1,1,1,4] => 1111000 => 7
[1,4,1,1] => [1,1,1,4] => [1,1,1,4] => 1111000 => 7
[4,1,1,1] => [1,1,1,4] => [1,1,1,4] => 1111000 => 7
[1,1,2,2] => [1,1,2,2] => [1,1,2,2] => 111010 => 6
[1,2,1,2] => [1,1,2,2] => [1,1,2,2] => 111010 => 6
[1,2,2,1] => [1,1,2,2] => [1,1,2,2] => 111010 => 6
[2,1,1,2] => [1,1,2,2] => [1,1,2,2] => 111010 => 6
[2,1,2,1] => [1,1,2,2] => [1,1,2,2] => 111010 => 6
[2,2,1,1] => [1,1,2,2] => [1,1,2,2] => 111010 => 6
[1,1,2,3] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[1,1,3,2] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[1,2,1,3] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[1,2,3,1] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[1,3,1,2] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[1,3,2,1] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[2,1,1,3] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[2,1,3,1] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[2,3,1,1] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[3,1,1,2] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[3,1,2,1] => [1,1,2,3] => [1,1,2,3] => 1110100 => 7
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[1,4,2,3] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[1,4,3,2] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[2,4,1,3] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[2,4,3,1] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[3,1,2,4] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[3,1,4,2] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[3,2,1,4] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[3,2,4,1] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[3,4,1,2] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[3,4,2,1] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[4,1,2,3] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[4,1,3,2] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[4,2,1,3] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[4,2,3,1] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[4,3,1,2] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[4,3,2,1] => [1,2,3,4] => [1,2,3,4] => 1101001000 => ? = 10
[1,1,1,3,5] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,1,1,5,3] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,1,3,1,5] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,1,3,5,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,1,5,1,3] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,1,5,3,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,3,1,1,5] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,3,1,5,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,3,5,1,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,5,1,1,3] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,5,1,3,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,5,3,1,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[3,1,1,1,5] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[3,1,1,5,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[3,1,5,1,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[3,5,1,1,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[5,1,1,1,3] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[5,1,1,3,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[5,1,3,1,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[5,3,1,1,1] => [1,1,1,3,5] => [1,1,1,3,5] => ? => ? = 11
[1,1,1,4,4] => [1,1,1,4,4] => [1,1,1,4,4] => ? => ? = 11
[1,1,4,1,4] => [1,1,1,4,4] => [1,1,1,4,4] => ? => ? = 11
[1,1,4,4,1] => [1,1,1,4,4] => [1,1,1,4,4] => ? => ? = 11
[1,4,1,1,4] => [1,1,1,4,4] => [1,1,1,4,4] => ? => ? = 11
[1,4,1,4,1] => [1,1,1,4,4] => [1,1,1,4,4] => ? => ? = 11
[1,4,4,1,1] => [1,1,1,4,4] => [1,1,1,4,4] => ? => ? = 11
Description
The flex of a binary word.
This is the product of the lex statistic ([[St001436]], augmented by 1) and its frequency ([[St000627]]), see [1, §8].
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000395The sum of the heights of the peaks of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001342The number of vertices in the center of a graph. St001875The number of simple modules with projective dimension at most 1. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000144The pyramid weight of the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000189The number of elements in the poset. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!