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Your data matches 89 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 2
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1

Description

The largest part of an integer partition.

Matching statistic: St000668

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 2
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 2
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 1

Description

The least common multiple of the parts of the partition.

Matching statistic: St000319

Mapping

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1 = 2 - 1
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 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

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1 = 2 - 1
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 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

Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[2,2]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1],[2],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2],[3],[4]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[2,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,2],[3,3]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,2],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,3],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1,1],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1,2],[2,2]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1],[2],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2],[3],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[3],[4],[5]]
 => [1,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,2],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,2],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,3],[3,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[2,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[3,3],[4,4]]
 => [2,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,1],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,2],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,4],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,3],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[2,4],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [2]
 => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 1 = 2 - 1
[[1,1,1],[2,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 1
[[1,1,1],[3,3]]
 => [3,2]
 => [2]
 => [1,1]
 => 0 = 1 - 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: St001864

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%

Values

[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,2]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1],[2],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,3]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[3,3]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,2],[2,3]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,3]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,2],[3,3]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,3],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1],[2],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[3],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[3,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,2],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,3],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,3],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,2],[3,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,2],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[2,3],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[3,3],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,1],[3],[4]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,4],[2],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[2,2],[3],[4]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[2,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 2
[[1,1,1],[2,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,1],[3,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,2],[2,3]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,3],[2,2]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,2],[3,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,3],[2,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,3],[3,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,2,2],[2,3]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,2,3],[2,3]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[2,2,2],[3,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[2,2,3],[3,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 2
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 2
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1
[[1,1,1,1],[2,2]]
 => [5,6,1,2,3,4] => [6,1,5,2,3,4] => [6,1,5,2,3,4] => ? = 1
[[1,1,1,2],[2,2]]
 => [4,5,1,2,3,6] => [5,1,4,2,3,6] => [5,1,4,2,3,6] => ? = 1
[[1,1,2,2],[2,2]]
 => [3,4,1,2,5,6] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 1
[[1,1,1],[2,2,2]]
 => [4,5,6,1,2,3] => [6,1,5,2,4,3] => [6,1,5,2,4,3] => ? = 3
[[1],[2],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[5],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1,1],[2,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,1],[3,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,1],[4,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,2],[2,4]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,4],[2,2]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,2],[3,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,3],[2,4]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,4],[2,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,2],[4,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,4],[2,4]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,3],[3,4]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,4],[3,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,3],[4,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,4],[3,4]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,4],[4,4]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,2,2],[2,4]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,2,3],[2,4]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,4],[2,3]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,2],[4,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,2,4],[2,4]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,4]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1

Description

The number of excedances of a signed permutation. For a signed permutation

$\pi\in\mathfrak H_n$ , this is

$\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$ .

Matching statistic: St001904

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001904: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%

Values

[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,2]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1],[2],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,3]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[3,3]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2],[2,3]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,3]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[3,3]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,3],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1],[2],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[3],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[3,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[4,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,3],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[4,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,3],[4,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[3,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[4,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[2,3],[4,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[3,3],[4,4]]
 => [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,1],[3],[4]]
 => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,4],[2],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[2,2],[3],[4]]
 => [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[2,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 2
[[1,1,1],[2,3]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,1],[3,3]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,2],[2,3]]
 => [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,3],[2,2]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,2],[3,3]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,3],[2,3]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,3],[3,3]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,2,2],[2,3]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,3]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,2,3],[2,3]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,3]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[2,2,2],[3,3]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[2,2,3],[3,3]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 2
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 2
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 1
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 1
[[1,1,1,1],[2,2]]
 => [5,6,1,2,3,4] => [6,5,2,3,4,1] => [6,5,2,3,4,1] => ? = 1
[[1,1,1,2],[2,2]]
 => [4,5,1,2,3,6] => [5,4,2,3,1,6] => [5,4,2,3,1,6] => ? = 1
[[1,1,2,2],[2,2]]
 => [3,4,1,2,5,6] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 1
[[1,1,1],[2,2,2]]
 => [4,5,6,1,2,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 3
[[1],[2],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[5],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1,1],[2,4]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,1],[3,4]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,1],[4,4]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,2],[2,4]]
 => [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,4],[2,2]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,2],[3,4]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,3],[2,4]]
 => [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,4],[2,3]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,2],[4,4]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,4],[2,4]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,3],[3,4]]
 => [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,4],[3,3]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,3],[4,4]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,4],[3,4]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,4],[4,4]]
 => [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,2,2],[2,4]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,4]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,2,3],[2,4]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,4],[2,3]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,2],[4,4]]
 => [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,2,4],[2,4]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,4]]
 => [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1

Description

The length of the initial strictly increasing segment of a parking function.

Matching statistic: St001862

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%

Values

[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,2]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1],[2],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,3]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[3,3]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,2],[2,3]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,3]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[2,2],[3,3]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,3],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1],[2],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[3],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[3,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,2],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,3],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[2,2],[3,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[2,2],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[2,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,3],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[3,3],[4,4]]
 => [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[1,1],[3],[4]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,4],[2],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[2,2],[3],[4]]
 => [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 1 = 2 - 1
[[1,1,1],[2,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,1],[3,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,2],[2,3]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[[1,1,3],[2,2]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,2],[3,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,3],[2,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,3],[3,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,2,2],[2,3]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,2,3],[2,3]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[2,2,2],[3,3]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[2,2,3],[3,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 2 - 1
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 2 - 1
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2 - 1
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2 - 1
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 1 - 1
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 1 - 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 1 - 1
[[1,1,1,1],[2,2]]
 => [5,6,1,2,3,4] => [6,1,5,2,3,4] => [6,1,5,2,3,4] => ? = 1 - 1
[[1,1,1,2],[2,2]]
 => [4,5,1,2,3,6] => [5,1,4,2,3,6] => [5,1,4,2,3,6] => ? = 1 - 1
[[1,1,2,2],[2,2]]
 => [3,4,1,2,5,6] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 1 - 1
[[1,1,1],[2,2,2]]
 => [4,5,6,1,2,3] => [6,1,5,2,4,3] => [6,1,5,2,4,3] => ? = 3 - 1
[[1],[2],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[5],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1,1],[2,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,1],[3,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,1],[4,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,2],[2,4]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[[1,1,4],[2,2]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,2],[3,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,3],[2,4]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[[1,1,4],[2,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,2],[4,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,4],[2,4]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,3],[3,4]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[[1,1,4],[3,3]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,3],[4,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,4],[3,4]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,4],[4,4]]
 => [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,2,2],[2,4]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,2,3],[2,4]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,4],[2,3]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,2],[4,4]]
 => [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,2,4],[2,4]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,4]]
 => [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1

Description

The number of crossings of a signed permutation. A crossing of a signed permutation

$\pi$ is a pair

$(i, j)$ of indices such that *

$i < j \leq \pi(i) < \pi(j)$ , or *

$-i < j \leq -\pi(i) < \pi(j)$ , or *

$i > j > \pi(i) > \pi(j)$ .

Matching statistic: St001882
(load all 3 compositions to match this statistic)

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%

Values

[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,2]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1],[2],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[4]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,3]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[3,3]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,2],[2,3]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,3]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,2],[3,3]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,3],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1],[2],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[3],[4],[5]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[3,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[4,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,2],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,3],[2,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[4,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,3],[4,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,2],[3,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,2],[4,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,3],[3,4]]
 => [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,3],[4,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[3,3],[4,4]]
 => [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,1],[3],[4]]
 => [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,4],[2],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[2,2],[3],[4]]
 => [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1,1,1],[2,3]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,1],[3,3]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,2],[2,3]]
 => [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[[1,1,3],[2,2]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,2],[3,3]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,3],[2,3]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,3],[3,3]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,2,2],[2,3]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,3]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,2,3],[2,3]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,3]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[2,2,2],[3,3]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[2,2,3],[3,3]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 - 1
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 - 1
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2 - 1
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2 - 1
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 1 - 1
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 1 - 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 1 - 1
[[1,1,1,1],[2,2]]
 => [5,6,1,2,3,4] => [6,5,1,2,3,4] => [6,5,1,2,3,4] => ? = 1 - 1
[[1,1,1,2],[2,2]]
 => [4,5,1,2,3,6] => [5,4,1,2,3,6] => [5,4,1,2,3,6] => ? = 1 - 1
[[1,1,2,2],[2,2]]
 => [3,4,1,2,5,6] => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 1 - 1
[[1,1,1],[2,2,2]]
 => [4,5,6,1,2,3] => [5,4,6,1,2,3] => [5,4,6,1,2,3] => ? = 3 - 1
[[1],[2],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[5],[6]]
 => [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1,1],[2,4]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,1],[3,4]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,1],[4,4]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,2],[2,4]]
 => [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[[1,1,4],[2,2]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,2],[3,4]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,3],[2,4]]
 => [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[[1,1,4],[2,3]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,2],[4,4]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,4],[2,4]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,3],[3,4]]
 => [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[[1,1,4],[3,3]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,3],[4,4]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,4],[3,4]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,4],[4,4]]
 => [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,2,2],[2,4]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,4]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,2,3],[2,4]]
 => [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,4],[2,3]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,2],[4,4]]
 => [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,2,4],[2,4]]
 => [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,4]]
 => [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1

Description

The number of occurrences of a type-B 231 pattern in a signed permutation. For a signed permutation

$\pi\in\mathfrak H_n$ , a triple

$-n \leq i < j < k\leq n$ is an occurrence of the type-B

$231$ pattern, if

$1 \leq j < k$ ,

$\pi(i) < \pi(j)$ and

$\pi(i)$ is one larger than

$\pi(k)$ , i.e.,

$\pi(i) = \pi(k) + 1$ if

$\pi(k) \neq -1$ and

$\pi(i) = 1$ otherwise.

Matching statistic: St000633

Mapping

Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 50%

Values

[[1],[2],[3]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1,1],[2,2]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1],[2],[4]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1],[3],[4]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[2],[3],[4]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1,1],[2,3]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,1],[3,3]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,2],[2,3]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[1,2],[3,3]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[2,2],[3,3]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,1],[2],[3]]
 => [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,2],[2],[3]]
 => [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,3],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,1,1],[2,2]]
 => [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1,1,2],[2,2]]
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1],[2],[5]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1],[3],[5]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1],[4],[5]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[2],[3],[5]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[2],[4],[5]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[3],[4],[5]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1,1],[2,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,1],[3,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,1],[4,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,2],[2,4]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[1,2],[3,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,3],[2,4]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[1,2],[4,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,3],[3,4]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[1,3],[4,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[2,2],[3,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[2,2],[4,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[2,3],[3,4]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[2,3],[4,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[3,3],[4,4]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,1],[2],[4]]
 => [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,1],[3],[4]]
 => [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,2],[2],[4]]
 => [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,2],[3],[4]]
 => [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,3],[2],[4]]
 => [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,4],[2],[4]]
 => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,3],[3],[4]]
 => [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[2,2],[3],[4]]
 => [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[2,3],[3],[4]]
 => [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[2,4],[3],[4]]
 => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
 => 2
[[1,1,1],[2,3]]
 => [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1,1,1],[3,3]]
 => [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1,1,2],[2,3]]
 => [3,5,1,2,4] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 1
[[1,1,3],[2,2]]
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1,1,2],[3,3]]
 => [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1,1,3],[2,3]]
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1,1,3],[3,3]]
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1,2,2],[2,3]]
 => [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 1
[[1,2,2],[3,3]]
 => [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1,2,3],[2,3]]
 => [2,4,1,3,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[[1,2,3],[3,3]]
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[2,2,2],[3,3]]
 => [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[2,2,3],[3,3]]
 => [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 1
[[1,1,1],[2],[3]]
 => [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
 => ? = 2
[[1,1,2],[2],[3]]
 => [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[[1,1,3],[2],[3]]
 => [4,3,1,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 2
[[1,2,2],[2],[3]]
 => [5,2,1,3,4] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[[1,2,3],[2],[3]]
 => [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
 => ? = 2
[[1,3,3],[2],[3]]
 => [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
 => ? = 2
[[1,1],[2,2],[3]]
 => [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
 => ? = 1
[[1,1],[2,3],[3]]
 => [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
 => ? = 1
[[1,2],[2,3],[3]]
 => [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
 => ? = 1
[[1,1,1,1],[2,2]]
 => [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
 => ? = 1
[[1,1,1,2],[2,2]]
 => [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
 => ? = 1
[[1,1,2,2],[2,2]]
 => [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
 => ? = 1
[[1,1,1],[2,2,2]]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
 => ? = 3
[[1],[2],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1],[3],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1],[4],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1],[5],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[2],[3],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[2],[4],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[2],[5],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[3],[4],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[3],[5],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[4],[5],[6]]
 => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[[1,1],[2,5]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,1],[3,5]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,1],[4,5]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,1],[5,5]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,2],[2,5]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[1,2],[3,5]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,3],[2,5]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[1,2],[4,5]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,4],[2,5]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[1,2],[5,5]]
 => [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ? = 1
[[1,3],[3,5]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[1,4],[3,5]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[1,4],[4,5]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[2,3],[3,5]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[2,4],[3,5]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1
[[2,4],[4,5]]
 => [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 1

Description

The size of the automorphism group of a poset. A poset automorphism is a permutation of the elements of the poset preserving the order relation.

The following 79 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001399The distinguishing number of a poset. St000850The number of 1/2-balanced pairs in a poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001754The number of tolerances of a finite lattice. St000640The rank of the largest boolean interval in a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000528The height of a poset. St000906The length of the shortest maximal chain in a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000084The number of subtrees. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000080The rank of the poset. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000522The number of 1-protected nodes of a rooted tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching. St001207The Lowey length of the algebra

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001625The Möbius invariant of a lattice. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000327The number of cover relations in a poset. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000635The number of strictly order preserving maps of a poset into itself. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001645The pebbling number of a connected graph. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001877Number of indecomposable injective modules with projective dimension 2. St001964The interval resolution global dimension of a poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. 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

$K[x]/(x^n)$ . St000189The number of elements in the poset. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000400The path length of an ordered tree. St000529The number of permutations whose descent word is the given binary word. St000180The number of chains of a poset. St000416The number of inequivalent increasing trees of an ordered tree. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St000100The number of linear extensions of a poset. St001909The number of interval-closed sets of a poset. St000410The tree factorial of an ordered tree. St000634The number of endomorphisms of a poset. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral.