- FindStat

Your data matches 24 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001123

Mapping

St001123: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2]
 => 1
[1,1]
 => 1
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 0
[3,1]
 => 1
[2,2]
 => 0
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 1
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 1
[3,3]
 => 0
[3,2,1]
 => 2
[3,1,1,1]
 => 1
[2,2,2]
 => 0
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 0
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 0
[4,3]
 => 0
[4,2,1]
 => 1
[4,1,1,1]
 => 1
[3,3,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 0
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 0
[5,3]
 => 0
[5,2,1]
 => 0
[5,1,1,1]
 => 1

Description

The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{21^{n-2}}$ , for

$\lambda\vdash n$ .

Matching statistic: St001139
(load all 2 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001139: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%

Values

[2]
 => []
 => []
 => ? ∊ {1,1}
[1,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1}
[3]
 => []
 => []
 => ? ∊ {0,0}
[2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0}
[1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[4]
 => []
 => []
 => ? ∊ {0,1}
[3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,1}
[2,2]
 => [2]
 => [1,0,1,0]
 => 0
[2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[5]
 => []
 => []
 => ? ∊ {0,1}
[4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,1}
[3,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[6]
 => []
 => []
 => ? ∊ {0,2}
[5,1]
 => [1]
 => [1,0]
 => ? ∊ {0,2}
[4,2]
 => [2]
 => [1,0,1,0]
 => 0
[4,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[7]
 => []
 => []
 => ? ∊ {1,1}
[6,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1}
[5,2]
 => [2]
 => [1,0,1,0]
 => 0
[5,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[4,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[4,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[3,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[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]
 => 0
[8]
 => []
 => []
 => ? ∊ {0,2}
[7,1]
 => [1]
 => [1,0]
 => ? ∊ {0,2}
[6,2]
 => [2]
 => [1,0,1,0]
 => 0
[6,1,1]
 => [1,1]
 => [1,1,0,0]
 => 1
[5,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[5,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[5,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[4,4]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
[4,3,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
[4,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => 0
[9]
 => []
 => []
 => ? ∊ {1,1,1}
[8,1]
 => [1]
 => [1,0]
 => ? ∊ {1,1,1}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {1,1,1}
[10]
 => []
 => []
 => ? ∊ {0,0,2,3}
[9,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,2,3}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,2,3}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,2,3}
[11]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,2}
[10,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,2}
[3,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,1,1,2}
[2,2,2,1,1,1,1,1]
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,1,1,2}
[2,2,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,1,1,2}
[2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,1,1,2}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,1,1,1,1,2}
[12]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[11,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[4,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[3,3,1,1,1,1,1,1]
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[3,2,2,1,1,1,1,1]
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[3,2,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[3,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[2,2,2,2,1,1,1,1]
 => [2,2,2,1,1,1,1]
 => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[2,2,2,1,1,1,1,1,1]
 => [2,2,1,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[2,2,1,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[2,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}
[1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,2,3}

Description

The number of occurrences of hills of size 2 in a Dyck path. A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.

Matching statistic: St001137

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 80%●distinct values known / distinct values provided: 50%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,1}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,1}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,2}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,2}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,2}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,2}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,0,1,0]
 => 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,1,1}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,1,1}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,1,1}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,1,1}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0,1,0]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,1,1,1,1}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0,1,0]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [1,1,0,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,0,1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0,1,0]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,1,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2,3}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2,3}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2,3}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,1,1,2}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2}
[2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,1,1,2}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {0,0,0,0,0,1,1,2}
[12]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,3}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,3}
[10,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,3}
[9,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,3}
[8,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,3}
[7,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,3}
[6,6]
 => [6]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,2,3}

Description

Number of simple modules that are 3-regular in the corresponding Nakayama algebra.

Matching statistic: St001175
(load all 3 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 79%●distinct values known / distinct values provided: 75%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1}
[1,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,1}
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1,1}
[2,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[2,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,1,1}
[1,1,1,1]
 => [1,1,1]
 => [2,1]
 => [1]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,1}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1,1,1}
[3,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[3,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,1,1,1}
[2,2,1]
 => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [2,1]
 => [1]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,1,1,1}
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,2}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,2}
[4,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[4,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,1,1,2}
[3,3]
 => [3]
 => [3]
 => []
 => ? ∊ {0,0,1,1,2}
[3,2,1]
 => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [2,1]
 => [1]
 => 0
[2,2,2]
 => [2,2]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,0,1,1,2}
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [4,1]
 => [1]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,1,1}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {1,1,1,1,1}
[5,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[5,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,1,1,1,1}
[4,3]
 => [3]
 => [3]
 => []
 => ? ∊ {1,1,1,1,1}
[4,2,1]
 => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [2,1]
 => [1]
 => 0
[3,3,1]
 => [3,1]
 => [3,1]
 => [1]
 => 0
[3,2,2]
 => [2,2]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {1,1,1,1,1}
[2,2,2,1]
 => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [4,1]
 => [1]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [4,2]
 => [2]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,1,2}
[7,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1,1,1,2}
[6,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[6,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,1,1,1,2}
[5,3]
 => [3]
 => [3]
 => []
 => ? ∊ {0,1,1,1,2}
[5,2,1]
 => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [2,1]
 => [1]
 => 0
[4,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[4,3,1]
 => [3,1]
 => [3,1]
 => [1]
 => 0
[4,2,2]
 => [2,2]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,1,1,1,2}
[3,3,2]
 => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [3,2]
 => [2]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [4,1]
 => [1]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [4,1,1]
 => [1,1]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [4,2]
 => [2]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [4,2,1]
 => [2,1]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,1,1,1,1}
[8,1]
 => [1]
 => [1]
 => []
 => ? ∊ {1,1,1,1,1,1,1}
[7,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[7,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {1,1,1,1,1,1,1}
[6,3]
 => [3]
 => [3]
 => []
 => ? ∊ {1,1,1,1,1,1,1}
[6,2,1]
 => [2,1]
 => [1,1,1]
 => [1,1]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [2,1]
 => [1]
 => 0
[5,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[5,3,1]
 => [3,1]
 => [3,1]
 => [1]
 => 0
[5,2,2]
 => [2,2]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {1,1,1,1,1,1,1}
[4,4,1]
 => [4,1]
 => [2,2,1]
 => [2,1]
 => 0
[4,3,2]
 => [3,2]
 => [3,1,1]
 => [1,1]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [3,2]
 => [2]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[3,3,3]
 => [3,3]
 => [6]
 => []
 => ? ∊ {1,1,1,1,1,1,1}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [8]
 => []
 => ? ∊ {1,1,1,1,1,1,1}
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,1,2,3}
[9,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,2,3}
[8,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,2,3}
[7,3]
 => [3]
 => [3]
 => []
 => ? ∊ {0,0,0,1,1,1,2,3}
[6,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,0,0,1,1,1,2,3}
[5,5]
 => [5]
 => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,2,3}
[4,3,3]
 => [3,3]
 => [6]
 => []
 => ? ∊ {0,0,0,1,1,1,2,3}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [8]
 => []
 => ? ∊ {0,0,0,1,1,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,1,1,1,1,2}
[10,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1,1,1,1,1,1,2}
[9,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,1,1,1,1,1,1,2}
[8,3]
 => [3]
 => [3]
 => []
 => ? ∊ {0,1,1,1,1,1,1,2}
[7,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,1,1,1,1,1,1,2}
[6,5]
 => [5]
 => [5]
 => []
 => ? ∊ {0,1,1,1,1,1,1,2}
[5,3,3]
 => [3,3]
 => [6]
 => []
 => ? ∊ {0,1,1,1,1,1,1,2}
[3,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [8]
 => []
 => ? ∊ {0,1,1,1,1,1,1,2}

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St001264

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001264: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 76%●distinct values known / distinct values provided: 75%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[2,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {1,1,1,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1,1,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {1,1,1,1}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {1,1,1,1}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => 0
[3,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,2}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,2}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,2}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,2}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,2}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => 0
[4,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [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]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,1,1,1}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,1,1,1,1,1}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,1,1,1,1,1}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => 0
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,1,1,1,1,1}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,1,1,1,1,1}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => 0
[5,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0]
 => 0
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,1,1,1,1,1}
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,2,3}
[6,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,2,3}
[5,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,2,3}
[2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,2,3}
[1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[10,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[9,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[8,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[7,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[6,5]
 => [5]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[3,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[2,2,2,1,1,1,1,1]
 => [2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[2,2,1,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1}
[12]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3}
[11,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3}

Description

The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra.

Matching statistic: St000205
(load all 3 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,1}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1}
[1,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1,1}
[2,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,1,1}
[2,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,1}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1,1,1}
[3,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,1,1,1}
[3,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[2,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,1,1,1}
[2,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,2}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,2}
[4,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,1,1,2}
[4,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[3,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[3,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,1,1,2}
[3,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[2,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,1,1,2}
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,1,1,1}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1,1,1,1,1}
[5,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,1,1,1,1,1}
[5,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[4,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[4,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,1,1,1,1,1}
[4,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,1,1,1,1,1}
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,1,1,1,1,1}
[2,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,1,2}
[7,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,2}
[6,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,2}
[6,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[5,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[5,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,0,1,1,1,2}
[5,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[4,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,1,1,1,2}
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[3,3,2]
 => [3,2]
 => [4,1]
 => [1]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,2}
[3,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {0,0,0,1,1,1,2}
[2,2,2,1,1]
 => [2,2,1,1]
 => [5,1]
 => [1]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,1,1,1,1,1}
[8,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1}
[7,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1}
[7,1,1]
 => [1,1]
 => [1,1]
 => [1]
 => 0
[6,3]
 => [3]
 => [2,1]
 => [1]
 => 0
[6,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1}
[6,1,1,1]
 => [1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[5,4]
 => [4]
 => [2,2]
 => [2]
 => 0
[5,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[5,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1}
[5,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,4,1]
 => [4,1]
 => [3,2]
 => [2]
 => 0
[4,3,2]
 => [3,2]
 => [4,1]
 => [1]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1}
[4,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[3,3,3]
 => [3,3]
 => [3,2,1]
 => [2,1]
 => 0
[3,3,2,1]
 => [3,2,1]
 => [3,3]
 => [3]
 => 0
[3,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1}
[2,2,2,2,1]
 => [2,2,2,1]
 => [7]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1}
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,1,1,2,3}
[9,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,1,2,3}
[8,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,1,2,3}
[7,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,0,1,1,1,1,2,3}
[6,2,2]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,1,1,1,1,2,3}
[5,2,2,1]
 => [2,2,1]
 => [5]
 => []
 => ? ∊ {0,0,0,1,1,1,1,2,3}
[4,2,2,2]
 => [2,2,2]
 => [6]
 => []
 => ? ∊ {0,0,0,1,1,1,1,2,3}
[3,2,2,2,1]
 => [2,2,2,1]
 => [7]
 => []
 => ? ∊ {0,0,0,1,1,1,1,2,3}
[2,2,2,2,2]
 => [2,2,2,2]
 => [8]
 => []
 => ? ∊ {0,0,0,1,1,1,1,2,3}
[11]
 => []
 => ?
 => ?
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
[10,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
[9,2]
 => [2]
 => [2]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,2}
[8,2,1]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,1,1,1,1,1,1,1,1,1,2}

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. Given

$\lambda$ count how many ''integer partitions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex.

Matching statistic: St000206
(load all 2 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 71%●distinct values known / distinct values provided: 50%

Values

[2]
 => []
 => []
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => [1]
 => []
 => ? ∊ {1,1}
[3]
 => []
 => []
 => ?
 => ? ∊ {0,0,1}
[2,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1}
[1,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,1}
[4]
 => []
 => []
 => ?
 => ? ∊ {0,0,1,1}
[3,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,1}
[2,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[2,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,1,1}
[1,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {0,0,1,1}
[5]
 => []
 => []
 => ?
 => ? ∊ {0,0,1,1,1}
[4,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1}
[3,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[3,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1}
[2,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 0
[2,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {0,0,1,1,1}
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,0,1,1,1}
[6]
 => []
 => []
 => ?
 => ? ∊ {0,0,0,1,1,2}
[5,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,2}
[4,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[4,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,2}
[3,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 0
[3,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 0
[3,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {0,0,0,1,1,2}
[2,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,0,0,1,1,2}
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {0,0,0,1,1,2}
[7]
 => []
 => []
 => ?
 => ? ∊ {0,0,1,1,1,1,1}
[6,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[5,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[5,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[4,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 0
[4,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 0
[4,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[3,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[3,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[8]
 => []
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,2}
[7,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[6,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[6,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[5,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 0
[5,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 0
[5,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[4,4]
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[4,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[4,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[3,3,2]
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[2,2,2,2]
 => [2,2,2]
 => [3,3]
 => [3]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [4,2]
 => [2]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[9]
 => []
 => []
 => ?
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[8,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[7,2]
 => [2]
 => [1,1]
 => [1]
 => 0
[7,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[6,3]
 => [3]
 => [1,1,1]
 => [1,1]
 => 0
[6,2,1]
 => [2,1]
 => [2,1]
 => [1]
 => 0
[6,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[5,4]
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[5,3,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 0
[5,2,2]
 => [2,2]
 => [2,2]
 => [2]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [3,1]
 => [1]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[4,4,1]
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[4,3,2]
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[3,3,3]
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 0
[3,3,2,1]
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 0
[3,3,1,1,1]
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 0
[3,2,2,2]
 => [2,2,2]
 => [3,3]
 => [3]
 => 0
[3,2,2,1,1]
 => [2,2,1,1]
 => [4,2]
 => [2]
 => 0
[3,2,1,1,1,1]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[2,2,2,2,1]
 => [2,2,2,1]
 => [4,3]
 => [3]
 => 0
[2,2,2,1,1,1]
 => [2,2,1,1,1]
 => [5,2]
 => [2]
 => 0
[2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [8]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[10]
 => []
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,1,1,2,3}
[9,1]
 => [1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,3}
[8,1,1]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,3}
[7,1,1,1]
 => [1,1,1]
 => [3]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,3}
[6,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,3}
[5,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => []
 => ? ∊ {0,0,0,0,1,1,1,1,2,3}

Description

Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. Given

$\lambda$ count how many ''integer compositions''

$w$ (weight) there are, such that

$P_{\lambda,w}$ is non-integral, i.e.,

$w$ such that the Gelfand-Tsetlin polytope

$P_{\lambda,w}$ has at least one non-integral vertex. See also [[St000205]]. Each value in this statistic is greater than or equal to corresponding value in [[St000205]].

Matching statistic: St000929
(load all 4 compositions to match this statistic)

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000929: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 71%●distinct values known / distinct values provided: 50%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,1}
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1}
[2,1,1]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,1,1}
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [2]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,1}
[3,1,1]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,1,1,1}
[2,2,1]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,1,1,1}
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [2]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,2}
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,1,2}
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [2]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [2]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,1,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1}
[5,1,1]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1}
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1}
[4,2,1]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1}
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [2]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,1}
[3,2,2]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [2]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [3]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [3,2]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,1,2}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,2}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,2}
[6,1,1]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,2}
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,2}
[5,2,1]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,2}
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [2]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,1,2}
[4,3,1]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,2}
[4,2,2]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [2]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [2]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [3]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [4]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [3,2]
 => 0
[1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [3,2,1]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[7,1,1]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[6,2,1]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [2]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[5,3,1]
 => [3,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[5,2,2]
 => [2,2]
 => [2]
 => [1,1]
 => 1
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [2]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [2,1]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,1,1,1,1}
[4,3,2]
 => [3,2]
 => [2]
 => [1,1]
 => 1
[4,3,1,1]
 => [3,1,1]
 => [1,1]
 => [2]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [2,1]
 => [3]
 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [2,1]
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 0
[3,3,3]
 => [3,3]
 => [3]
 => [1,1,1]
 => 1
[3,3,2,1]
 => [3,2,1]
 => [2,1]
 => [3]
 => 0
[3,3,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => [2,1]
 => 0
[3,2,2,2]
 => [2,2,2]
 => [2,2]
 => [4]
 => 0
[3,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [2,2]
 => 0
[3,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [3,1]
 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [3,2]
 => 0
[2,2,2,2,1]
 => [2,2,2,1]
 => [2,2,1]
 => [2,2,1]
 => 0
[2,2,2,1,1,1]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [3,1,1]
 => 0
[2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [3,2]
 => 0
[2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [3,2,1]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,2,3}
[8,1,1]
 => [1,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,2,3}
[7,2,1]
 => [2,1]
 => [1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,2,3}

Description

The constant term of the character polynomial of an integer partition. The definition of the character polynomial can be found in [1]. Indeed, this constant term is

$0$ for partitions

$\lambda \neq 1^n$ and

$1$ for

$\lambda = 1^n$ .

Matching statistic: St000980

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 69% ●values known / values provided: 69%●distinct values known / distinct values provided: 75%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1}
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1}
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,1}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1}
[2,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1}
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1,2}
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1,2}
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,1,1,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1}
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1}
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1}
[3,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,1,1,1,2}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,1,1,1,2}
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,1,1,1,2}
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,1,1,1,2}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,1,1,1,2}
[4,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [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]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[5,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1,1,1}
[4,3,2]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,3,3]
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[3,3,2,1]
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[3,3,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 0
[3,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[3,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[2,2,2,2,1]
 => [2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,2,2,1,1,1]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1,1]
 => [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]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,3}
[8,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,3}
[7,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,1,1,2,3}

Description

The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. For example, the path

$111011010000$ has three peaks in positions

$03, 15, 26$ . The boxes below

$03$ are

$01,02,\textbf{12}$ , the boxes below

$15$ are

$\textbf{12},13,14,\textbf{23},\textbf{24},\textbf{34}$ , and the boxes below

$26$ are

$\textbf{23},\textbf{24},25,\textbf{34},35,45$ . We thus obtain the four boxes in positions

$12,23,24,34$ that are below at least two peaks.

Matching statistic: St001141

Mapping

Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001141: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 69%●distinct values known / distinct values provided: 50%

Values

[2]
 => []
 => ?
 => ?
 => ? ∊ {1,1}
[1,1]
 => [1]
 => []
 => []
 => ? ∊ {1,1}
[3]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1}
[2,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1}
[1,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1}
[4]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1}
[3,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1}
[2,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1}
[2,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1}
[1,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,1}
[4,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,1}
[3,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,1}
[3,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1}
[2,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1}
[2,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[6]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,2}
[5,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2}
[4,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2}
[4,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1,2}
[3,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1,2}
[3,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1,2}
[3,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[7]
 => []
 => ?
 => ?
 => ? ∊ {0,0,1,1,1,1,1}
[6,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[5,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[5,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1}
[4,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,1,1,1,1,1}
[4,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1}
[4,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,1,1,1,1,1}
[3,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[2,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[8]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,1,1,2}
[7,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2}
[6,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2}
[6,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,2}
[5,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2}
[5,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,2}
[5,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,0,0,1,1,2}
[4,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,1,1,2}
[4,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[4,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,3,2]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[3,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[3,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[3,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[2,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[2,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,1,1,1,1]
 => [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]
 => 0
[9]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,1,1,1,1,1,1}
[8,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1}
[7,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1}
[7,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1}
[6,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1}
[6,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1}
[6,1,1,1]
 => [1,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,4]
 => [4]
 => []
 => []
 => ? ∊ {0,0,0,1,1,1,1,1,1}
[5,3,1]
 => [3,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1}
[5,2,2]
 => [2,2]
 => [2]
 => [1,0,1,0]
 => 0
[5,2,1,1]
 => [2,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[5,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[4,4,1]
 => [4,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,1,1,1,1,1,1}
[4,3,2]
 => [3,2]
 => [2]
 => [1,0,1,0]
 => 0
[4,3,1,1]
 => [3,1,1]
 => [1,1]
 => [1,1,0,0]
 => 0
[4,2,2,1]
 => [2,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[4,2,1,1,1]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[4,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,3,3]
 => [3,3]
 => [3]
 => [1,0,1,0,1,0]
 => 0
[3,3,2,1]
 => [3,2,1]
 => [2,1]
 => [1,0,1,1,0,0]
 => 0
[3,3,1,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 0
[3,2,2,2]
 => [2,2,2]
 => [2,2]
 => [1,1,1,0,0,0]
 => 1
[3,2,2,1,1]
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 0
[3,2,1,1,1,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 0
[3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[2,2,2,2,1]
 => [2,2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,2,2,1,1,1]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 0
[2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[2,1,1,1,1,1,1,1]
 => [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]
 => 0
[10]
 => []
 => ?
 => ?
 => ? ∊ {0,0,0,0,0,0,0,1,1,2,3}
[9,1]
 => [1]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,2,3}
[8,2]
 => [2]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,2,3}
[8,1,1]
 => [1,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,2,3}
[7,3]
 => [3]
 => []
 => []
 => ? ∊ {0,0,0,0,0,0,0,1,1,2,3}
[7,2,1]
 => [2,1]
 => [1]
 => [1,0]
 => ? ∊ {0,0,0,0,0,0,0,1,1,2,3}

Description

The number of occurrences of hills of size 3 in a Dyck path. A hill of size three is a subpath beginning at height zero, consisting of three up steps followed by three down steps.

The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000124The cardinality of the preimage of the Simion-Schmidt map. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001498The normalised height of a Nakayama algebra with magnitude 1. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000338The number of pixed points of a permutation. St000649The number of 3-excedences of a permutation. St001617The dimension of the space of valuations of a lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000326The position of the first one in a binary word after appending a 1 at the end. St000455The second largest eigenvalue of a graph if it is integral.