Your data matches 60 different statistics following compositions of up to 3 maps.
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Matching statistic: St000987
(load all 22 compositions to match this statistic)
Mapping
St000987: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => 0 = 1 - 1
([],2)
 => 0 = 1 - 1
([(0,1)],2)
 => 1 = 2 - 1
([],3)
 => 0 = 1 - 1
([(1,2)],3)
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
([],4)
 => 0 = 1 - 1
([(2,3)],4)
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
([],5)
 => 0 = 1 - 1
([(3,4)],5)
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(1,4),(2,3)],5)
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => 4 = 5 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 4 = 5 - 1
Description
The number of positive eigenvalues of the Laplacian matrix of the graph. This is the number of vertices minus the number of connected components of the graph.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1]]
 => 1
([],2)
 => [1,1]
 => [[1],[2]]
 => 1
([(0,1)],2)
 => [2]
 => [[1,2]]
 => 2
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => 1
([(1,2)],3)
 => [2,1]
 => [[1,3],[2]]
 => 2
([(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => 3
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => 3
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 2
([(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,3,4],[2]]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => 4
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 2
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 4
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,5],[3,4]]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => 5
Description
The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St000377
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => 0 = 1 - 1
([],2)
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,1)],2)
 => [2]
 => [1,1]
 => 1 = 2 - 1
([],3)
 => [1,1,1]
 => [2,1]
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1]
 => [3]
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 2 = 3 - 1
([],4)
 => [1,1,1,1]
 => [3,1]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [2,2]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [4]
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([],5)
 => [1,1,1,1,1]
 => [3,2]
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [3,1,1]
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [2,2,1]
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [5]
 => 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [4,1]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [5]
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St001176
(load all 2 compositions to match this statistic)
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => 0 = 1 - 1
([],2)
 => [1,1]
 => [2]
 => 0 = 1 - 1
([(0,1)],2)
 => [2]
 => [1,1]
 => 1 = 2 - 1
([],3)
 => [1,1,1]
 => [3]
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1]
 => [2,1]
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => 2 = 3 - 1
([],4)
 => [1,1,1,1]
 => [4]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => 3 = 4 - 1
([],5)
 => [1,1,1,1,1]
 => [5]
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => 4 = 5 - 1
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000026
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1,0]
 => [1,0]
 => 1
([],2)
 => [2] => [1,1,0,0]
 => [1,0,1,0]
 => 1
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 2
([],3)
 => [3] => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 1
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 3
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 3
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 2
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 5
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 5
Description
The position of the first return of a Dyck path.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => ([],1)
 => [1]
 => 1
([],2)
 => [2] => ([],2)
 => [1,1]
 => 1
([(0,1)],2)
 => [1,1] => ([(0,1)],2)
 => [2]
 => 2
([],3)
 => [3] => ([],3)
 => [1,1,1]
 => 1
([(1,2)],3)
 => [1,2] => ([(1,2)],3)
 => [2,1]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([(0,1),(0,2),(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => [4] => ([],4)
 => [1,1,1,1]
 => 1
([(2,3)],4)
 => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
([(1,3),(2,3)],4)
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,3),(1,2)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(1,2),(1,3),(2,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 4
([],5)
 => [5] => ([],5)
 => [1,1,1,1,1]
 => 1
([(3,4)],5)
 => [1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 2
([(2,4),(3,4)],5)
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,4),(2,3)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(2,3),(2,4),(3,4)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
Description
The largest part of an integer partition.
Matching statistic: St000738
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [1]
 => [[1]]
 => 1
([],2)
 => [1,1]
 => [2]
 => [[1,2]]
 => 1
([(0,1)],2)
 => [2]
 => [1,1]
 => [[1],[2]]
 => 2
([],3)
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 1
([(1,2)],3)
 => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 2
([(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3
([],4)
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 1
([(2,3)],4)
 => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 2
([(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
([(0,3),(1,2)],4)
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4
([],5)
 => [1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 1
([(3,4)],5)
 => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 2
([(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(1,4),(2,3)],5)
 => [2,2,1]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St000839
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => [1,0]
 => {{1}}
 => 1
([],2)
 => [2] => [1,1,0,0]
 => {{1,2}}
 => 1
([(0,1)],2)
 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 2
([],3)
 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 1
([(1,2)],3)
 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => [1,0,1,0,1,0]
 => {{1},{2},{3}}
 => 3
([(0,1),(0,2),(1,2)],3)
 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 3
([],4)
 => [4] => [1,1,1,1,0,0,0,0]
 => {{1,2,3,4}}
 => 1
([(2,3)],4)
 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
([(1,3),(2,3)],4)
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 4
([(0,3),(1,2)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 3
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4
([(1,2),(1,3),(2,3)],4)
 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4}}
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 4
([],5)
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => {{1,2,3,4,5}}
 => 1
([(3,4)],5)
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 2
([(2,4),(3,4)],5)
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 5
([(1,4),(2,3)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 4
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 4
([(2,3),(2,4),(3,4)],5)
 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 5
Description
The largest opener of a set partition. An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.
Matching statistic: St001389
Mapping
Mp00152: Graphs —Laplacian multiplicities⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1] => ([],1)
 => [1]
 => 1
([],2)
 => [2] => ([],2)
 => [1,1]
 => 1
([(0,1)],2)
 => [1,1] => ([(0,1)],2)
 => [2]
 => 2
([],3)
 => [3] => ([],3)
 => [1,1,1]
 => 1
([(1,2)],3)
 => [1,2] => ([(1,2)],3)
 => [2,1]
 => 2
([(0,2),(1,2)],3)
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 3
([(0,1),(0,2),(1,2)],3)
 => [2,1] => ([(0,2),(1,2)],3)
 => [3]
 => 3
([],4)
 => [4] => ([],4)
 => [1,1,1,1]
 => 1
([(2,3)],4)
 => [1,3] => ([(2,3)],4)
 => [2,1,1]
 => 2
([(1,3),(2,3)],4)
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 3
([(0,3),(1,3),(2,3)],4)
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,3),(1,2)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
([(0,3),(1,2),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(1,2),(1,3),(2,3)],4)
 => [2,2] => ([(1,3),(2,3)],4)
 => [3,1]
 => 3
([(0,3),(1,2),(1,3),(2,3)],4)
 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,2),(0,3),(1,2),(1,3)],4)
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 4
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => [4]
 => 4
([],5)
 => [5] => ([],5)
 => [1,1,1,1,1]
 => 1
([(3,4)],5)
 => [1,4] => ([(3,4)],5)
 => [2,1,1,1]
 => 2
([(2,4),(3,4)],5)
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(1,4),(2,4),(3,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,4),(2,4),(3,4)],5)
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,4),(2,3)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(1,4),(2,3),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,1),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(2,3),(2,4),(3,4)],5)
 => [2,3] => ([(2,4),(3,4)],5)
 => [3,1,1]
 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,3),(1,4),(2,3),(2,4)],5)
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,3),(2,3),(2,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,1),(2,3),(2,4),(3,4)],5)
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => 4
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 5
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
Matching statistic: St000074
Mapping
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000074: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],1)
 => [1]
 => [[1]]
 => [[1]]
 => 0 = 1 - 1
([],2)
 => [1,1]
 => [[1],[2]]
 => [[1,1],[1]]
 => 0 = 1 - 1
([(0,1)],2)
 => [2]
 => [[1,2]]
 => [[2,0],[1]]
 => 1 = 2 - 1
([],3)
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => 0 = 1 - 1
([(1,2)],3)
 => [2,1]
 => [[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => 1 = 2 - 1
([(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
 => [3]
 => [[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => 2 = 3 - 1
([],4)
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,1,1,1],[1,1,1],[1,1],[1]]
 => 0 = 1 - 1
([(2,3)],4)
 => [2,1,1]
 => [[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1 = 2 - 1
([(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,3),(1,2)],4)
 => [2,2]
 => [[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => [[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2 = 3 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => [[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([],5)
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => 0 = 1 - 1
([(3,4)],5)
 => [2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => 1 = 2 - 1
([(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(1,4),(2,3)],5)
 => [2,2,1]
 => [[1,2],[3,4],[5]]
 => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
 => [3,1,1]
 => [[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
 => [3,2]
 => [[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,1]
 => [[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 3 = 4 - 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [5]
 => [[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => 4 = 5 - 1
Description
The number of special entries. An entry $a_{i,j}$ of a Gelfand-Tsetlin pattern is special if $a_{i-1,j-i} > a_{i,j} > a_{i-1,j}$. That is, it is neither boxed nor circled.
The following 50 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000157The number of descents of a standard tableau. St000211The rank of the set partition. St000228The size of a partition. St000245The number of ascents of a permutation. St000293The number of inversions of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000441The number of successions of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000672The number of minimal elements in Bruhat order not less than the permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St000171The degree of the graph. St000369The dinv deficit of a Dyck path. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St000727The largest label of a leaf in the binary search tree associated with the permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St000054The first entry of the permutation. St000141The maximum drop size of a permutation. St001725The harmonious chromatic number of a graph. St001330The hat guessing number of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000653The last descent of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000051The size of the left subtree of a binary tree. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000840The number of closers smaller than the largest opener in a perfect matching. St001812The biclique partition number of a graph. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. St000080The rank of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000526The number of posets with combinatorially isomorphic order polytopes. St000680The Grundy value for Hackendot on posets. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset.