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Mp00323: Integer partitions Loehr-Warrington inverseInteger partitions
St001587: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 0
[2,1]
=> [1,1,1]
=> 0
[2,2]
=> [2,1,1]
=> 1
[3,1,1]
=> [2,1,1,1]
=> 1
[3,2,1]
=> [1,1,1,1,1,1]
=> 0
[4,3]
=> [2,2,1,1,1]
=> 1
[4,1,1,1]
=> [2,2,2,1]
=> 1
[2,2,2,1]
=> [3,2,2]
=> 1
[3,3,2]
=> [2,2,1,1,1,1]
=> 1
[5,1,1,1,1]
=> [3,2,2,2]
=> 1
[5,3,1,1]
=> [2,2,1,1,1,1,1,1]
=> 1
[4,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> 0
[4,2,2,1,1]
=> [5,2,1,1,1]
=> 1
[4,4,3,2]
=> [2,2,2,1,1,1,1,1,1,1]
=> 1
[5,4,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 0
Description
Half of the largest even part of an integer partition. The largest even part is recorded by [[St000995]].
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00138: Dyck paths to noncrossing partitionSet partitions
St000253: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> {{1,5},{2},{3,4},{6}}
=> 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> {{1},{2,6},{3},{4,5}}
=> 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5},{6}}
=> 0
Description
The crossing number of a set partition. This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00118: Dyck paths swap returns and last descentDyck paths
St001011: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St001174: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [6,4,2,3,1,5] => 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [5,3,4,2,6,1] => 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => 0
Description
The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00099: Dyck paths bounce pathDyck paths
St001418: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St001507: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
Description
The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St001761: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
Description
The maximal multiplicity of a letter in a reduced word of a permutation. For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00242: Dyck paths Hessenberg posetPosets
St000298: Posets ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 2 = 1 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2 = 1 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 2 = 1 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> ([(0,3),(0,4),(2,5),(3,5),(4,1),(4,2)],6)
=> 2 = 1 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> 2 = 1 + 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 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.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
St000451: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 1 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 2 = 1 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,3,5,1,6] => 2 = 1 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => 2 = 1 + 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2 = 1 + 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1 = 0 + 1
Description
The length of the longest pattern of the form k 1 2...(k-1).
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
St000862: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 2 = 1 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2 = 1 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 1 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => 2 = 1 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,4,1,5,3,6] => 2 = 1 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,2,6,4] => 2 = 1 + 1
[4,4,3,2]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2 = 1 + 1
[5,4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1 = 0 + 1
Description
The number of parts of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the number of parts of the shifted shape.
The following 433 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000920The logarithmic height of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000183The side length of the Durfee square of an integer partition. St000254The nesting number of a set partition. St000272The treewidth of a graph. St000292The number of ascents of a binary word. St000306The bounce count of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000392The length of the longest run of ones in a binary word. St000455The second largest eigenvalue of a graph if it is integral. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000628The balance of a binary word. St000640The rank of the largest boolean interval in a poset. St000651The maximal size of a rise in a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000665The number of rafts of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000710The number of big deficiencies of a permutation. St000730The maximal arc length of a set partition. St000845The maximal number of elements covered by an element in a poset. St000864The number of circled entries of the shifted recording tableau of a permutation. St000884The number of isolated descents of a permutation. St000919The number of maximal left branches of a binary tree. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001353The number of prime nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001592The maximal number of simple paths between any two different vertices of a graph. St001665The number of pure excedances of a permutation. St001673The degree of asymmetry of an integer composition. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001928The number of non-overlapping descents in a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000007The number of saliances of the permutation. St000010The length of the partition. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000092The number of outer peaks of a permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000099The number of valleys of a permutation, including the boundary. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000259The diameter of a connected graph. St000308The height of the tree associated to a permutation. St000325The width of the tree associated to a permutation. St000346The number of coarsenings of a partition. St000396The register function (or Horton-Strahler number) of a binary tree. St000397The Strahler number of a rooted tree. St000470The number of runs in a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000485The length of the longest cycle of a permutation. St000527The width of the poset. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000668The least common multiple of the parts of the partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000822The Hadwiger number of the graph. St000903The number of different parts of an integer composition. St000904The maximal number of repetitions of an integer composition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001096The size of the overlap set of a permutation. 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. 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: St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001280The number of parts of an integer partition that are at least two. St001330The hat guessing number of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001432The order dimension of the partition. St001494The Alon-Tarsi number of a graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001530The depth of a Dyck path. St001580The acyclic chromatic number of a graph. St001734The lettericity of a graph. St001735The number of permutations with the same set of runs. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000172The Grundy number of a graph. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000891The number of distinct diagonal sums of a permutation matrix. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001093The detour number of a graph. St001116The game chromatic number of a graph. St001963The tree-depth of a graph. St000251The number of nonsingleton blocks of a set partition. St000779The tier of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000402Half the size of the symmetry class of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000143The largest repeated part of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000013The height of a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St001394The genus of a permutation. St000846The maximal number of elements covering an element of a poset. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001733The number of weak left to right maxima of a Dyck path. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000661The number of rises of length 3 of a Dyck path. St000761The number of ascents in an integer composition. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000872The number of very big descents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000617The number of global maxima of a Dyck path. St000624The normalized sum of the minimal distances to a greater element. St000647The number of big descents of a permutation. St000905The number of different multiplicities of parts of an integer composition. St000996The number of exclusive left-to-right maxima of a permutation. St000444The length of the maximal rise of a Dyck path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. 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. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001489The maximum of the number of descents and the number of inverse descents. St000386The number of factors DDU in a Dyck path. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000667The greatest common divisor of the parts of the partition. St000711The number of big exceedences of a permutation. St000731The number of double exceedences of a permutation. St000781The number of proper colouring schemes of a Ferrers diagram. St000993The multiplicity of the largest part of an integer partition. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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 $eA$. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001549The number of restricted non-inversions between exceedances. St001557The number of inversions of the second entry of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000314The number of left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000478Another weight of a partition according to Alladi. St000672The number of minimal elements in Bruhat order not less than the permutation. St000767The number of runs in an integer composition. St000929The constant term of the character polynomial of an integer partition. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001175The size of a partition minus the hook length of the base cell. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. St001298The number of repeated entries in the Lehmer code of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001555The order of a signed permutation. St001569The maximal modular displacement of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. 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). St000638The number of up-down runs of a permutation. St000670The reversal length of a permutation. 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 $eA$. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001760The number of prefix or suffix reversals needed to sort a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000454The largest eigenvalue of a graph if it is integral. St000618The number of self-evacuating tableaux of given shape. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for 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. St000944The 3-degree of an integer partition. 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. St001586The number of odd parts smaller than the largest even part in an integer partition. St000091The descent variation of a composition. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000461The rix statistic of a permutation. St000534The number of 2-rises of a permutation. St000836The number of descents of distance 2 of a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001728The number of invisible descents of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St000153The number of adjacent cycles of a permutation. St000335The difference of lower and upper interactions. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001516The number of cyclic bonds of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001435The number of missing boxes in the first row. St001487The number of inner corners of a skew partition. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001948The number of augmented double ascents of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001052The length of the exterior of a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000962The 3-shifted major index of a permutation. St001556The number of inversions of the third entry of a permutation. St000141The maximum drop size of a permutation. St000352The Elizalde-Pak rank of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St000054The first entry of the permutation. St000264The girth of a graph, which is not a tree. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001488The number of corners of a skew partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000260The radius of a connected graph. St000355The number of occurrences of the pattern 21-3. St000407The number of occurrences of the pattern 2143 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000516The number of stretching pairs of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000709The number of occurrences of 14-2-3 or 14-3-2. St000807The sum of the heights of the valleys of the associated bargraph. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001712The number of natural descents of a standard Young tableau. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000241The number of cyclical small excedances. St000287The number of connected components of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000805The number of peaks of the associated bargraph. St000808The number of up steps of the associated bargraph. St000958The number of Bruhat factorizations of a permutation. St000983The length of the longest alternating subword. St001184Number of indecomposable injective modules with grade at least 1 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 St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001778The largest greatest common divisor of an element and its image in a permutation. St001884The number of borders of a binary word. St000381The largest part of an integer composition. St000829The Ulam distance of a permutation to the identity permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001415The length of the longest palindromic prefix of a binary word. St001589The nesting number of a perfect matching. St000735The last entry on the main diagonal of a standard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St000488The number of cycles of a permutation of length at most 2. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000873The aix statistic of a permutation. St001438The number of missing boxes of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001684The reduced word complexity of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000302The determinant of the distance matrix of a connected graph. St000456The monochromatic index of a connected graph. St000538The number of even inversions of a permutation. St000837The number of ascents of distance 2 of a permutation. St000909The number of maximal chains of maximal size in a poset. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St001820The size of the image of the pop stack sorting operator. St000075The orbit size of a standard tableau under promotion. St000489The number of cycles of a permutation of length at most 3. St000679The pruning number of an ordered tree. St001060The distinguishing index of a graph. St001285The number of primes in the column sums of the two line notation of a permutation. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001439The number of even weak deficiencies and of odd weak exceedences. St000348The non-inversion sum of a binary word. 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. St000632The jump number of the poset. St000682The Grundy value of Welter's game on a binary word. St000881The number of short braid edges in the graph of braid moves of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001730The number of times the path corresponding to a binary word crosses the base line. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001857The number of edges in the reduced word graph of a signed permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000079The number of alternating sign matrices for a given Dyck path. St000307The number of rowmotion orbits of a poset. St000315The number of isolated vertices of a graph. St000327The number of cover relations in a poset. St000390The number of runs of ones in a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001008Number of indecomposable injective modules with projective dimension 1 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. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001490The number of connected components of a skew partition. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001668The number of points of the poset minus the width of the poset. St001768The number of reduced words of a signed permutation. St001863The number of weak excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000297The number of leading ones in a binary word. St000657The smallest part of an integer composition. St000876The number of factors in the Catalan decomposition of a binary word. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000982The length of the longest constant subword. St001570The minimal number of edges to add to make a graph Hamiltonian. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001637The number of (upper) dissectors of a poset. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001875The number of simple modules with projective dimension at most 1.