searching the database
Your data matches 713 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: St000071
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> 1
Description
The number of maximal chains in a poset.
Matching statistic: St000173
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000173: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000173: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [[1,2]]
=> 1
[[2,2]]
=> [[2,2]]
=> 1
[[1],[2]]
=> [[1,2]]
=> 1
[[1,3]]
=> [[1,3]]
=> 1
[[2,3]]
=> [[2,3]]
=> 2
[[3,3]]
=> [[3,3]]
=> 1
[[1],[3]]
=> [[1,3]]
=> 1
[[2],[3]]
=> [[2,3]]
=> 2
[[1,1,2]]
=> [[1,1,2]]
=> 1
[[1,2,2]]
=> [[1,2,2]]
=> 1
[[2,2,2]]
=> [[2,2,2]]
=> 1
[[1,1],[2]]
=> [[1,1,2]]
=> 1
[[1,2],[2]]
=> [[1,2,2]]
=> 1
[[1,1,1,2]]
=> [[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> [[1,1,2,2]]
=> 1
[[1,2,2,2]]
=> [[1,2,2,2]]
=> 1
[[2,2,2,2]]
=> [[2,2,2,2]]
=> 1
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> 1
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> 1
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> 1
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> 1
[[1,1,1,1,2]]
=> [[1,1,1,1,2]]
=> 1
[[1,1,1,2,2]]
=> [[1,1,1,2,2]]
=> 1
[[1,1,2,2,2]]
=> [[1,1,2,2,2]]
=> 1
[[1,2,2,2,2]]
=> [[1,2,2,2,2]]
=> 1
[[2,2,2,2,2]]
=> [[2,2,2,2,2]]
=> 1
[[1,1,1,1],[2]]
=> [[1,1,1,1,2]]
=> 1
[[1,1,1,2],[2]]
=> [[1,1,1,2,2]]
=> 1
[[1,1,2,2],[2]]
=> [[1,1,2,2,2]]
=> 1
[[1,2,2,2],[2]]
=> [[1,2,2,2,2]]
=> 1
[[1,1,1],[2,2]]
=> [[1,1,1,2,2]]
=> 1
[[1,1,2],[2,2]]
=> [[1,1,2,2,2]]
=> 1
[[1,1,1,1,1,2]]
=> [[1,1,1,1,1,2]]
=> 1
[[1,1,1,1,2,2]]
=> [[1,1,1,1,2,2]]
=> 1
[[1,1,1,2,2,2]]
=> [[1,1,1,2,2,2]]
=> 1
[[1,1,2,2,2,2]]
=> [[1,1,2,2,2,2]]
=> 1
[[1,2,2,2,2,2]]
=> [[1,2,2,2,2,2]]
=> 1
[[2,2,2,2,2,2]]
=> [[2,2,2,2,2,2]]
=> 1
[[1,1,1,1,1],[2]]
=> [[1,1,1,1,1,2]]
=> 1
[[1,1,1,1,2],[2]]
=> [[1,1,1,1,2,2]]
=> 1
[[1,1,1,2,2],[2]]
=> [[1,1,1,2,2,2]]
=> 1
[[1,1,2,2,2],[2]]
=> [[1,1,2,2,2,2]]
=> 1
[[1,2,2,2,2],[2]]
=> [[1,2,2,2,2,2]]
=> 1
[[1,1,1,1],[2,2]]
=> [[1,1,1,1,2,2]]
=> 1
[[1,1,1,2],[2,2]]
=> [[1,1,1,2,2,2]]
=> 1
[[1,1,2,2],[2,2]]
=> [[1,1,2,2,2,2]]
=> 1
[[1,1,1],[2,2,2]]
=> [[1,1,1,2,2,2]]
=> 1
Description
The segment statistic of a semistandard tableau.
Let ''T'' be a tableau. A ''k''-segment of ''T'' (in the ''i''th row) is defined to be a maximal consecutive sequence of ''k''-boxes in the ith row. Note that the possible ''i''-boxes in the ''i''th row are not considered to be ''i''-segments. Then seg(''T'') is the total number of ''k''-segments in ''T'' as ''k'' varies over all possible values.
Matching statistic: St000298
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000307
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000307: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000307: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> 1
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St000527
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000527: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> 1
Description
The width of the poset.
This is the size of the poset's longest antichain, also called Dilworth number.
Matching statistic: St000909
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000909: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000909: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1],[2]]
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1],[2]]
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> 1
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000632
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000632: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
[[1],[3]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1],[2,2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[1,1,1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1,1],[2,2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> 0 = 1 - 1
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St001633
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001633: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 2 - 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 2 - 1
[[1],[3]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1],[2,2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[1,1,1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1,1],[2,2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 1 - 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 1 - 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> 0 = 1 - 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 1 - 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> 0 = 1 - 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> 0 = 1 - 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> 0 = 1 - 1
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00110: Posets —Greene-Kleitman invariant⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [4,1]
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [5,1]
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> [2]
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [6]
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [7]
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [4]
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [5]
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> [2]
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> [3]
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> [1]
=> 1
Description
The length of the partition.
Matching statistic: St000088
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00214: Semistandard tableaux —subcrystal⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000088: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000088: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 2
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 2
[[1],[3]]
=> ([(0,1)],2)
=> [1]
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 1
[[1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,2],[2,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,1,1,1,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1,1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> [1]
=> 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> [1]
=> 1
[[1,1,1,1,1],[2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,1,2],[2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1,2,2,2],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> [1]
=> 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> [1]
=> 1
[[1,1,1,1],[2,2]]
=> ([],1)
=> [1]
=> 1
[[1,1,1,2],[2,2]]
=> ([(0,1)],2)
=> [1]
=> 1
[[1,1,2,2],[2,2]]
=> ([(0,2),(2,1)],3)
=> [1]
=> 1
[[1,1,1],[2,2,2]]
=> ([],1)
=> [1]
=> 1
Description
The row sums of the character table of the symmetric group.
Equivalently, this is the multiplicity of the irreducible representation corresponding to the given partition in the adjoint representation of the symmetric group.
The following 703 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000146The Andrews-Garvan crank of a partition. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000172The Grundy number of a graph. St000179The product of the hook lengths of the integer partition. St000183The side length of the Durfee square of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000228The size of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000269The number of acyclic orientations of a graph. St000270The number of forests contained in a graph. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000346The number of coarsenings of a partition. St000363The number of minimal vertex covers of a graph. St000378The diagonal inversion number of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. St000453The number of distinct Laplacian eigenvalues of a graph. 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. St000468The Hosoya index of a graph. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000482The (zero)-forcing number of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000544The cop number of a graph. St000644The number of graphs with given frequency partition. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St000784The maximum of the length and the largest part of the integer partition. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000935The number of ordered refinements of an integer partition. St000972The composition number of a graph. St000992The alternating sum of the parts of an integer partition. St001029The size of the core of a graph. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001261The Castelnuovo-Mumford regularity of a graph. St001280The number of parts of an integer partition that are at least two. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001360The number of covering relations in Young's lattice below a partition. St001367The smallest number which does not occur as degree of a vertex in a graph. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001474The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001484The number of singletons of an integer partition. St001494The Alon-Tarsi number of a graph. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001624The breadth of a lattice. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001725The harmonious chromatic number of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001883The mutual visibility number of a graph. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001963The tree-depth of a graph. St001985The sum of the greatest common divisors of all subsets of the parts of an integer partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000081The number of edges of a graph. St000108The number of partitions contained in the given partition. St000142The number of even parts of a partition. St000145The Dyson rank of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000171The degree of the graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000185The weighted size of a partition. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000263The Szeged index of a graph. St000265The Wiener index of a graph. St000271The chromatic index of a graph. St000272The treewidth of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000361The second Zagreb index of a graph. St000362The size of a minimal vertex cover of a graph. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000387The matching number of a graph. St000454The largest eigenvalue of a graph if it is integral. St000481The number of upper covers of a partition in dominance order. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000532The total number of rook placements on a Ferrers board. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000547The number of even non-empty partial sums of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001071The beta invariant of the graph. St001092The number of distinct even parts of a partition. St001120The length of a longest path in a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001341The number of edges in the center of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001393The induced matching number of a graph. St001400The total number of Littlewood-Richardson tableaux of given shape. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001525The number of symmetric hooks on the diagonal of a partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001638The book thickness of a graph. St001644The dimension of a graph. St001657The number of twos in an integer partition. St001736The total number of cycles in a graph. St001743The discrepancy of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001792The arboricity of a graph. St001814The number of partitions interlacing the given partition. St001826The maximal number of leaves on a vertex of a graph. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001962The proper pathwidth of a graph. St001971The number of negative eigenvalues of the adjacency matrix of the graph. St001973The Gromov width of a graph. St001982The number of orbits of the action of a permutation of given cycle type on the set of edges of the complete graph. St000013The height of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000093The cardinality of a maximal independent set of vertices of a graph. St000288The number of ones in a binary word. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000758The length of the longest staircase fitting into an integer composition. St000759The smallest missing part in an integer partition. St000764The number of strong records in an integer composition. St000765The number of weak records in an integer composition. St000767The number of runs in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000820The number of compositions obtained by rotating the composition. St000897The number of different multiplicities of parts of an integer partition. St000903The number of different parts of an integer composition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001057The Grundy value of the game of creating an independent set in a graph. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001732The number of peaks visible from the left. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001984A recursive count of subwords of the form 01, 10 and 11. St000008The major index of the composition. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000059The inversion number of a standard tableau as defined by Haglund and Stevens. St000148The number of odd parts of a partition. St000157The number of descents of a standard tableau. St000160The multiplicity of the smallest part of a partition. St000169The cocharge of a standard tableau. St000225Difference between largest and smallest parts in a partition. St000292The number of ascents of a binary word. St000330The (standard) major index of a standard tableau. St000336The leg major index of a standard tableau. St000340The number of non-final maximal constant sub-paths of length greater than one. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000448The number of pairs of vertices of a graph with distance 2. St000475The number of parts equal to 1 in a partition. St000548The number of different non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000749The 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. St000761The number of ascents in an integer composition. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St000983The length of the longest alternating subword. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001127The sum of the squares of the parts of a partition. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001310The number of induced diamond graphs in a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001673The degree of asymmetry of an integer composition. St001697The shifted natural comajor index of a standard Young tableau. St001712The number of natural descents of a standard Young tableau. St001799The number of proper separations of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001779The order of promotion on the set of linear extensions of a poset. St001397Number of pairs of incomparable elements in a finite poset. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St000343The number of spanning subgraphs of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001734The lettericity of a graph. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001649The length of a longest trail in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St001869The maximum cut size of a graph. St000053The number of valleys of the Dyck path. St000087The number of induced subgraphs. St000244The cardinality of the automorphism group of a graph. St000258The burning number of a graph. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000286The number of connected components of the complement of a graph. St000335The difference of lower and upper interactions. St000364The exponent of the automorphism group of a graph. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000469The distinguishing number of a graph. St000479The Ramsey number of a graph. St000636The hull number of a graph. St000675The number of centered multitunnels of a Dyck path. St000722The number of different neighbourhoods in a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000926The clique-coclique number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001109The number of proper colourings of a graph with as few colours as possible. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001315The dissociation number of a graph. St001316The domatic number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001642The Prague dimension of a graph. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001672The restrained domination number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St001757The number of orbits of toric promotion on a graph. St001758The number of orbits of promotion on a graph. St001802The number of endomorphisms of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000466The Gutman (or modified Schultz) index of a connected graph. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001271The competition number of a graph. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001323The independence gap of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001345The Hamming dimension of a graph. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001352The number of internal nodes in the modular decomposition of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001362The normalized Knill dimension of a graph. St001391The disjunction number of a graph. St001395The number of strictly unfriendly partitions of a graph. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001578The minimal number of edges to add or remove to make a graph a line graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001783The number of odd automorphisms of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001827The number of two-component spanning forests of a graph. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001949The rigidity index of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001983The number of indecomposable injective modules that are pure. St001964The interval resolution global dimension of a poset. St000100The number of linear extensions of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001877Number of indecomposable injective modules with projective dimension 2. St000331The number of upper interactions of a Dyck path. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001780The order of promotion on the set of standard tableaux of given shape. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000015The number of peaks of a Dyck path. St000117The number of centered tunnels of a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001530The depth of a Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001738The minimal order of a graph which is not an induced subgraph of the given graph. 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. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000736The last entry in the first row of a semistandard tableau. St001933The largest multiplicity of a part in an integer partition. St000570The Edelman-Greene number of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000219The number of occurrences of the pattern 231 in a permutation. St000462The major index minus the number of excedences of a permutation. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000842The breadth of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000045The number of linear extensions of a binary tree. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000402Half the size of the symmetry class of a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000530The number of permutations with the same descent word as the given permutation. St000619The number of cyclic descents of a permutation. St000652The maximal difference between successive positions of a permutation. St000690The size of the conjugacy class of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000847The number of standard Young tableaux whose descent set is the binary word. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St001081The number of minimal length factorizations of a permutation into star transpositions. St001220The width of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001570The minimal number of edges to add to make a graph Hamiltonian. St001722The number of minimal chains with small intervals between a binary word and the top element. St001890The maximum magnitude of the Möbius function of a poset. St000216The absolute length of a permutation. St000226The convexity of a permutation. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000376The bounce deficit of a Dyck path. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000624The normalized sum of the minimal distances to a greater element. St000629The defect of a binary word. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000653The last descent of a permutation. St000673The number of non-fixed points of a permutation. St000682The Grundy value of Welter's game on a binary word. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000794The mak of a permutation. St000795The mad of a permutation. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000809The reduced reflection length of the permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000830The total displacement of a permutation. St000831The number of indices that are either descents or recoils. St000833The comajor index of a permutation. St000836The number of descents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St000921The number of internal inversions of a binary word. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001060The distinguishing index of a graph. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001082The number of boxed occurrences of 123 in a permutation. St001114The number of odd descents of a permutation. St001130The number of two successive successions in a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001388The number of non-attacking neighbors of a permutation. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001731The factorization defect of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001976The bin statistic of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000181The number of connected components of the Hasse diagram for the poset. St000418The number of Dyck paths that are weakly below a Dyck path. St000444The length of the maximal rise of a Dyck path. St000635The number of strictly order preserving maps of a poset into itself. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000914The sum of the values of the Möbius function of a poset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000101The cocharge of a semistandard tableau. St000478Another weight of a partition according to Alladi. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000003The number of standard Young tableaux of the partition. St000026The position of the first return of a Dyck path. St000048The multinomial of the parts of a partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000075The orbit size of a standard tableau under promotion. St000079The number of alternating sign matrices for a given Dyck path. St000182The number of permutations whose cycle type is the given integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000391The sum of the positions of the ones in a binary word. St000392The length of the longest run of ones in a binary word. St000393The number of strictly increasing runs in a binary word. St000517The Kreweras number of an integer partition. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000627The exponent of a binary word. St000628The balance of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000631The number of distinct palindromic decompositions of a binary word. St000655The length of the minimal rise of a Dyck path. St000691The number of changes of a binary word. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000712The number of semistandard Young tableau of given shape, with entries at most 4. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000753The Grundy value for the game of Kayles on a binary word. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000792The Grundy value for the game of ruler on a binary word. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000913The number of ways to refine the partition into singletons. St000922The minimal number such that all substrings of this length are unique. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000982The length of the longest constant subword. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001118The acyclic chromatic index of a graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001129The product of the squares of the parts of a partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001267The length of the Lyndon factorization of the binary word. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001437The flex of a binary word. St001481The minimal height of a peak of a Dyck path. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001568The smallest positive integer that does not appear twice in the partition. St001595The number of standard Young tableaux of the skew partition. St001597The Frobenius rank of a skew partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001809The index of the step at the first peak of maximal height in a Dyck path. St001838The number of nonempty primitive factors of a binary word. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000934The 2-degree of an integer partition. St000735The last entry on the main diagonal of a standard tableau. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000782The indicator function of whether a given perfect matching is an L & P matching. 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. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition.
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!