Your data matches 91 different statistics following compositions of up to 3 maps.
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Matching statistic: St000877
(load all 3 compositions to match this statistic)
Mapping
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 0
00 => 2
01 => 1
10 => 0
11 => 0
000 => 3
001 => 2
010 => 1
011 => 1
100 => 1
101 => 0
110 => 0
111 => 0
0000 => 4
0001 => 3
0010 => 2
0011 => 2
0100 => 2
0101 => 1
0110 => 1
0111 => 1
1000 => 2
1001 => 1
1010 => 0
1011 => 0
1100 => 0
1101 => 0
1110 => 0
1111 => 0
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St001570
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => ([(0,2),(1,2)],3)
 => 1
1 => [1,1] => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
00 => [3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 2
01 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
10 => [1,2] => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
000 => [4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
010 => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
100 => [1,3] => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
0000 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 4
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
0100 => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
1000 => [1,4] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => 3
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
1111 => [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)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
Description
The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
0 => [2] => [[2],[]]
 => []
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => []
 => ? ∊ {0,1,2}
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 0
10 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,1,2}
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,1,2}
000 => [4] => [[4],[]]
 => []
 => ? ∊ {1,1,2,3}
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
100 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {1,1,2,3}
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
0000 => [5] => [[5],[]]
 => []
 => ? ∊ {2,2,2,3,4}
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 2
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
1000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
0 => [2] => [[2],[]]
 => []
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => []
 => ? ∊ {0,1,2}
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 0
10 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,1,2}
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,1,2}
000 => [4] => [[4],[]]
 => []
 => ? ∊ {1,1,2,3}
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
100 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {1,1,2,3}
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
0000 => [5] => [[5],[]]
 => []
 => ? ∊ {2,2,2,3,4}
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 2
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
1000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
Description
The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001176
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
0 => [2] => [[2],[]]
 => []
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => []
 => ? ∊ {0,1,2}
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 0
10 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,1,2}
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,1,2}
000 => [4] => [[4],[]]
 => []
 => ? ∊ {1,1,2,3}
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 0
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 1
100 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {1,1,2,3}
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
0000 => [5] => [[5],[]]
 => []
 => ? ∊ {1,2,2,3,4}
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 0
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 0
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 1
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 2
1000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {1,2,2,3,4}
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 1
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {1,2,2,3,4}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {1,2,2,3,4}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {1,2,2,3,4}
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: St001279
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Values
0 => [2] => [[2],[]]
 => []
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => []
 => ? ∊ {0,1,2}
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 0
10 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,1,2}
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,1,2}
000 => [4] => [[4],[]]
 => []
 => ? ∊ {1,1,1,3}
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 2
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
100 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {1,1,1,3}
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {1,1,1,3}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {1,1,1,3}
0000 => [5] => [[5],[]]
 => []
 => ? ∊ {1,1,1,1,2}
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 3
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 4
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
1000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {1,1,1,1,2}
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 2
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {1,1,1,1,2}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {1,1,1,1,2}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {1,1,1,1,2}
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
0 => [2] => [[2],[]]
 => []
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => []
 => ? ∊ {0,1,2}
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 0
10 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,1,2}
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,1,2}
000 => [4] => [[4],[]]
 => []
 => ? ∊ {1,1,2,3}
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
100 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {1,1,2,3}
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
0000 => [5] => [[5],[]]
 => []
 => ? ∊ {2,2,2,3,4}
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 1
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
1000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001525
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001525: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 60%
Values
0 => [2] => [[2],[]]
 => []
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,2}
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 1
10 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,2}
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,2}
000 => [4] => [[4],[]]
 => []
 => ? ∊ {0,1,2,3}
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 0
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
100 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,1,2,3}
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,1,2,3}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,1,2,3}
0000 => [5] => [[5],[]]
 => []
 => ? ∊ {2,2,2,3,4}
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 0
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 0
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
1000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 0
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {2,2,2,3,4}
Description
The number of symmetric hooks on the diagonal of a partition.
Matching statistic: St001541
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001541: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [[2],[]]
 => []
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => []
 => ? ∊ {0,1,2}
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 0
10 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,1,2}
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,1,2}
000 => [4] => [[4],[]]
 => []
 => ? ∊ {1,1,2,3}
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 1
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 0
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
100 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {1,1,2,3}
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {1,1,2,3}
0000 => [5] => [[5],[]]
 => []
 => ? ∊ {1,1,2,2,2}
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 3
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 4
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 0
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 2
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
1000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {1,1,2,2,2}
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {1,1,2,2,2}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {1,1,2,2,2}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {1,1,2,2,2}
Description
The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.
Matching statistic: St001606
(load all 2 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Values
0 => [2] => [[2],[]]
 => []
 => ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
 => []
 => ? ∊ {0,1}
00 => [3] => [[3],[]]
 => []
 => ? ∊ {0,0,2}
01 => [2,1] => [[2,2],[1]]
 => [1]
 => 1
10 => [1,2] => [[2,1],[]]
 => []
 => ? ∊ {0,0,2}
11 => [1,1,1] => [[1,1,1],[]]
 => []
 => ? ∊ {0,0,2}
000 => [4] => [[4],[]]
 => []
 => ? ∊ {0,0,1,3}
001 => [3,1] => [[3,3],[2]]
 => [2]
 => 2
010 => [2,2] => [[3,2],[1]]
 => [1]
 => 1
011 => [2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
100 => [1,3] => [[3,1],[]]
 => []
 => ? ∊ {0,0,1,3}
101 => [1,2,1] => [[2,2,1],[1]]
 => [1]
 => 1
110 => [1,1,2] => [[2,1,1],[]]
 => []
 => ? ∊ {0,0,1,3}
111 => [1,1,1,1] => [[1,1,1,1],[]]
 => []
 => ? ∊ {0,0,1,3}
0000 => [5] => [[5],[]]
 => []
 => ? ∊ {0,0,0,2,4}
0001 => [4,1] => [[4,4],[3]]
 => [3]
 => 3
0010 => [3,2] => [[4,3],[2]]
 => [2]
 => 2
0011 => [3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 2
0100 => [2,3] => [[4,2],[1]]
 => [1]
 => 1
0101 => [2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
1000 => [1,4] => [[4,1],[]]
 => []
 => ? ∊ {0,0,0,2,4}
1001 => [1,3,1] => [[3,3,1],[2]]
 => [2]
 => 2
1010 => [1,2,2] => [[3,2,1],[1]]
 => [1]
 => 1
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
1100 => [1,1,3] => [[3,1,1],[]]
 => []
 => ? ∊ {0,0,0,2,4}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 1
1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,2,4}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => []
 => ? ∊ {0,0,0,2,4}
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions.
The following 81 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St000699The toughness times the least common multiple of 1,. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001651The Frankl number of a lattice. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001330The hat guessing number of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000455The second largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000940The number of characters of the symmetric group whose value on the partition is zero. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.