- FindStat

Your data matches 82 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000147

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => 3

Description

The largest part of an integer partition.

Matching statistic: St000668

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 2
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 2
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 2
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => 3

Description

The least common multiple of the parts of the partition.

Matching statistic: St000319

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 2 = 3 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 2 = 3 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0 = 1 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => 1 = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => 2 = 3 - 1

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 2 = 3 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 2 = 3 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0 = 1 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => 1 = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => 2 = 3 - 1

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

Matching statistic: St001918

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => 1 = 2 - 1
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => 1 = 2 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => 1 = 2 - 1
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 2 = 3 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => 1 = 2 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => 2 = 3 - 1
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => 0 = 1 - 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => 2 = 3 - 1
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => 1 = 2 - 1
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0 = 1 - 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0 = 1 - 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 1 = 2 - 1
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => 1 = 2 - 1
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 0 = 1 - 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => 1 = 2 - 1
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => 1 = 2 - 1
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => 0 = 1 - 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => 2 = 3 - 1

Description

The degree of the cyclic sieving polynomial corresponding to an integer partition. Let

$\lambda$ be an integer partition of

$n$ and let

$N$ be the least common multiple of the parts of

$\lambda$ . Fix an arbitrary permutation

$\pi$ of cycle type

$\lambda$ . Then

$\pi$ induces a cyclic action of order

$N$ on

$\{1,\dots,n\}$ . The corresponding character can be identified with the cyclic sieving polynomial

$C_\lambda(q)$ of this action, modulo

$q^N-1$ . Explicitly, it is

$\sum_{p\in\lambda} [p]_{q^{N/p}},$ where

$[p]_q = 1+\dots+q^{p-1}$ is the

$q$ -integer. This statistic records the degree of

$C_\lambda(q)$ . Equivalently, it equals

$\left(1 - \frac{1}{\lambda_1}\right) N,$ where

$\lambda_1$ is the largest part of

$\lambda$ . The statistic is undefined for the empty partition.

Matching statistic: St000010
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => [2]
 => 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [3]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [2]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => [1,1]
 => 2
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [2,1]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [2]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,1]
 => 2
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [2,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [2,1]
 => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [2]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [1,1,1]
 => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,1]
 => 2
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [3]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [2]
 => 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [3]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [3,1]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [2]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [2,1]
 => 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [3]
 => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => [1,1]
 => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [2]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [3]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [2,1,1]
 => 3

Description

The length of the partition.

Matching statistic: St000676

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 2
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => [1,0,1,0]
 => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength

$n$ with

$k$ up steps in odd positions and

$k$ returns to the main diagonal are counted by the binomial coefficient

$\binom{n-1}{k-1}$ [3,4].

Matching statistic: St000734

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => [[1,2]]
 => 2
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [[1,2]]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [[1,2]]
 => 2
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [[1,2]]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [[1,2,3]]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => [[1,2]]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [[1,2]]
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => [[1,2]]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [[1,2,3]]
 => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [[1,2]]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [[1,2,3]]
 => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [[1,2]]
 => 2
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [[1,2],[3],[4]]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [[1,2],[3]]
 => 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => [[1,2]]
 => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [[1],[2]]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [[1,2,3],[4]]
 => 3

Description

The last entry in the first row of a standard tableau.

Matching statistic: St001039
(load all 2 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([],4)
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 2
([],5)
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 2
([],6)
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2]
 => [1,0,1,0]
 => 2
([],7)
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2]
 => [1,0,1,0]
 => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 3

Description

The maximal height of a column in the parallelogram polyomino associated with a Dyck path.

Matching statistic: St001280
(load all 5 compositions to match this statistic)

Mapping

Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],3)
 => [1,1,1]
 => [3]
 => 1
([],4)
 => [1,1,1,1]
 => [4]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 2
([],5)
 => [1,1,1,1,1]
 => [5]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 2
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 2
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 2
([],6)
 => [1,1,1,1,1,1]
 => [6]
 => 1
([(4,5)],6)
 => [2,1,1,1,1]
 => [5,1]
 => 1
([(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => 1
([(2,5),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 1
([(2,5),(3,4)],6)
 => [2,2,1,1]
 => [4,2]
 => 2
([(2,5),(3,4),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 1
([(1,2),(3,5),(4,5)],6)
 => [3,2,1]
 => [3,2,1]
 => 2
([(3,4),(3,5),(4,5)],6)
 => [3,1,1,1]
 => [4,1,1]
 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
 => [3,3]
 => [2,2,2]
 => 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 1
([(0,5),(1,4),(2,3)],6)
 => [2,2,2]
 => [3,3]
 => 2
([(0,1),(2,5),(3,4),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
 => [3,2,1]
 => [3,2,1]
 => 2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)
 => [3,3]
 => [2,2,2]
 => 3
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,1,1]
 => [3,1,1,1]
 => 1
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)
 => [3,3]
 => [2,2,2]
 => 3
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,2]
 => [2,2,1,1]
 => 2
([],7)
 => [1,1,1,1,1,1,1]
 => [7]
 => 1
([(5,6)],7)
 => [2,1,1,1,1,1]
 => [6,1]
 => 1
([(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [5,1,1]
 => 1
([(3,6),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [4,1,1,1]
 => 1
([(2,6),(3,6),(4,6),(5,6)],7)
 => [5,1,1]
 => [3,1,1,1,1]
 => 1
([(3,6),(4,5)],7)
 => [2,2,1,1,1]
 => [5,2]
 => 2
([(3,6),(4,5),(5,6)],7)
 => [4,1,1,1]
 => [4,1,1,1]
 => 1
([(2,3),(4,6),(5,6)],7)
 => [3,2,1,1]
 => [4,2,1]
 => 2
([(4,5),(4,6),(5,6)],7)
 => [3,1,1,1,1]
 => [5,1,1]
 => 1
([(2,6),(3,6),(4,5),(5,6)],7)
 => [5,1,1]
 => [3,1,1,1,1]
 => 1
([(1,2),(3,6),(4,6),(5,6)],7)
 => [4,2,1]
 => [3,2,1,1]
 => 2
([(3,6),(4,5),(4,6),(5,6)],7)
 => [4,1,1,1]
 => [4,1,1,1]
 => 1
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
 => [5,2]
 => [2,2,1,1,1]
 => 2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => [5,1,1]
 => [3,1,1,1,1]
 => 1
([(3,5),(3,6),(4,5),(4,6)],7)
 => [4,1,1,1]
 => [4,1,1,1]
 => 1
([(1,6),(2,6),(3,5),(4,5)],7)
 => [3,3,1]
 => [3,2,2]
 => 3
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
 => [15,1,1]
 => [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
 => ? = 1

Description

The number of parts of an integer partition that are at least two.

The following 72 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000451The length of the longest pattern of the form k 1 2. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000378The diagonal inversion number of an integer partition. St000306The bounce count of a Dyck path. St000439The position of the first down step of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra 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: St001733The number of weak left to right maxima of a Dyck path. St000053The number of valleys of the Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n-1}]$ by adding

$c_0$ to

$c_{n-1}$ . St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000678The number of up steps after the last double rise of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000443The number of long tunnels of a Dyck path. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001202Call 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. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000331The number of upper interactions of a Dyck path. 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. St001185The number of indecomposable injective modules of grade at least 2 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

$eA$ . 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001557The number of inversions of the second entry of a permutation. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . 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

$eA$ . St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St000934The 2-degree of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001271The competition number of a graph.