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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000319
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
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
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001918
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1]
=> 0
[-,+] => [1,2] => [1,1]
=> [1]
=> 0
[+,-] => [1,2] => [1,1]
=> [1]
=> 0
[-,-] => [1,2] => [1,1]
=> [1]
=> 0
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 0
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 0
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 0
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 0
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 0
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 0
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St000147
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1]
=> 1 = 0 + 1
[-,+] => [1,2] => [1,1]
=> [1]
=> 1 = 0 + 1
[+,-] => [1,2] => [1,1]
=> [1]
=> 1 = 0 + 1
[-,-] => [1,2] => [1,1]
=> [1]
=> 1 = 0 + 1
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 1 = 0 + 1
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 1 = 0 + 1
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 1 = 0 + 1
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
Description
The largest part of an integer partition.
Matching statistic: St001280
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[-,+] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[+,-] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[-,-] => [1,2] => [1,1]
=> [2]
=> 1 = 0 + 1
[+,+,+] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[-,+,+] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[+,-,+] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[-,-,+] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[-,+,-] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[+,-,-] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,1,1]
=> [3]
=> 1 = 0 + 1
[+,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[-,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[2,1,+] => [2,1,3] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[3,+,1] => [3,2,1] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [2,1]
=> [2,1]
=> 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [4]
=> 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [2,1,1]
=> 1 = 0 + 1
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [3,1]
=> 1 = 0 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000668
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 83% ●values known / values provided: 83%●distinct values known / distinct values provided: 100%
Values
[+,+] => [1,2] => [1,1]
=> [1]
=> ? = 0 + 1
[-,+] => [1,2] => [1,1]
=> [1]
=> ? = 0 + 1
[+,-] => [1,2] => [1,1]
=> [1]
=> ? = 0 + 1
[-,-] => [1,2] => [1,1]
=> [1]
=> ? = 0 + 1
[+,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,+] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 0 + 1
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? = 0 + 1
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> ? = 0 + 1
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> ? = 0 + 1
[3,+,1] => [3,2,1] => [2,1]
=> [1]
=> ? = 0 + 1
[3,-,1] => [3,2,1] => [2,1]
=> [1]
=> ? = 0 + 1
[+,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,+,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,-,+,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,+,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,+,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,-,-,+] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,-,+,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,+,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,+,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,-,4,3] => [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,3,2,+] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,3,2,-] => [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 0 + 1
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? = 0 + 1
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 0 + 1
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? = 0 + 1
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,+,+] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,-,+] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,+,-] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,-,-] => [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2 = 1 + 1
[2,3,1,+] => [2,3,1,4] => [3,1]
=> [1]
=> ? = 0 + 1
[2,3,1,-] => [2,3,1,4] => [3,1]
=> [1]
=> ? = 0 + 1
[2,4,+,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 0 + 1
[2,4,-,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? = 0 + 1
[3,1,2,+] => [3,1,2,4] => [3,1]
=> [1]
=> ? = 0 + 1
[3,1,2,-] => [3,1,2,4] => [3,1]
=> [1]
=> ? = 0 + 1
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[3,+,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? = 0 + 1
[3,-,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? = 0 + 1
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2 = 1 + 1
[4,1,+,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? = 0 + 1
[4,1,-,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? = 0 + 1
[4,+,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? = 0 + 1
[4,-,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? = 0 + 1
[4,+,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,-,+,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,+,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[4,-,-,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[+,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? = 0 + 1
[-,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? = 0 + 1
[+,3,5,2,4] => [1,3,5,2,4] => [4,1]
=> [1]
=> ? = 0 + 1
[-,3,5,2,4] => [1,3,5,2,4] => [4,1]
=> [1]
=> ? = 0 + 1
[+,4,2,5,3] => [1,4,2,5,3] => [4,1]
=> [1]
=> ? = 0 + 1
[-,4,2,5,3] => [1,4,2,5,3] => [4,1]
=> [1]
=> ? = 0 + 1
[+,4,5,3,2] => [1,4,5,3,2] => [4,1]
=> [1]
=> ? = 0 + 1
[-,4,5,3,2] => [1,4,5,3,2] => [4,1]
=> [1]
=> ? = 0 + 1
[+,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? = 0 + 1
[-,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? = 0 + 1
[+,5,4,2,3] => [1,5,4,2,3] => [4,1]
=> [1]
=> ? = 0 + 1
[-,5,4,2,3] => [1,5,4,2,3] => [4,1]
=> [1]
=> ? = 0 + 1
[2,3,4,1,+] => [2,3,4,1,5] => [4,1]
=> [1]
=> ? = 0 + 1
[2,3,4,1,-] => [2,3,4,1,5] => [4,1]
=> [1]
=> ? = 0 + 1
[2,3,5,+,1] => [2,3,5,4,1] => [4,1]
=> [1]
=> ? = 0 + 1
[2,3,5,-,1] => [2,3,5,4,1] => [4,1]
=> [1]
=> ? = 0 + 1
[2,4,1,3,+] => [2,4,1,3,5] => [4,1]
=> [1]
=> ? = 0 + 1
[2,4,1,3,-] => [2,4,1,3,5] => [4,1]
=> [1]
=> ? = 0 + 1
[2,4,+,5,1] => [2,4,3,5,1] => [4,1]
=> [1]
=> ? = 0 + 1
[2,4,-,5,1] => [2,4,3,5,1] => [4,1]
=> [1]
=> ? = 0 + 1
[2,5,1,+,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 0 + 1
[2,5,1,-,3] => [2,5,1,4,3] => [4,1]
=> [1]
=> ? = 0 + 1
[2,5,+,1,4] => [2,5,3,1,4] => [4,1]
=> [1]
=> ? = 0 + 1
[2,5,-,1,4] => [2,5,3,1,4] => [4,1]
=> [1]
=> ? = 0 + 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St001964
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[+,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 0
[-,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 0
[+,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 0
[-,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 0
[+,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[+,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[+,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[+,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[-,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[+,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[-,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,+] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,1,-] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,+,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[3,-,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[+,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[-,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[+,4,2,3] => [1,4,2,3] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[-,4,2,3] => [1,4,2,3] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[+,4,+,2] => [1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[-,4,+,2] => [1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[+,4,-,2] => [1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[-,4,-,2] => [1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,-,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,+,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1
[2,3,1,+] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,3,1,-] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[2,4,+,1] => [2,4,3,1] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[2,4,-,1] => [2,4,3,1] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,1,2,+] => [3,1,2,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,1,2,-] => [3,1,2,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,+,1,+] => [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,-,1,+] => [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,+,1,-] => [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,-,1,-] => [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0
[3,+,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,-,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 0
[3,4,1,2] => [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1
[4,1,+,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,1,-,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,+,1,3] => [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0
[4,-,1,3] => [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0
[4,+,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,-,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,+,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,-,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[-,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,-,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,-,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,-,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[+,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[+,-,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[+,+,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-,-,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-,+,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[+,-,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[-,-,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
[+,+,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Matching statistic: St000260
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[+,+] => [1,2] => [2] => ([],2)
=> ? = 0 + 1
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[+,-] => [1,2] => [2] => ([],2)
=> ? = 0 + 1
[-,-] => [1,2] => [2] => ([],2)
=> ? = 0 + 1
[+,+,+] => [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[+,-,-] => [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[-,-,-] => [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[+,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[-,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,-,1] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[-,-,-,+] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[+,-,-,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[+,+,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,+,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,-,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,-,4,3] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[+,3,2,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,3,2,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,3,2,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,3,2,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[+,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,3,4,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[+,4,2,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,4,2,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,4,+,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,4,+,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,4,-,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,4,-,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[2,1,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,1,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,3,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,3,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[2,4,+,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,4,-,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[3,1,2,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,1,2,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[3,+,1,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,-,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[3,+,1,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[3,-,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[3,+,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[4,1,+,2] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,+,1,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,+,+,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,-,+,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,-,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,+,-,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,+,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,4,3,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,5,+,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,3,2,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,4,2,3,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,4,+,2,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,5,2,+,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,5,+,2,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,5,+,+,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,1,2,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,+,1,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,2,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,+,2,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,+,1,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,+,+,1,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,2,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,+,2,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,+,+,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,+,1,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,+,1,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,+,+,1,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,+,+,+,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[+,-,+,+,+,+] => [1,3,4,5,6,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000456
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[+,+] => [1,2] => [2] => ([],2)
=> ? = 0 + 1
[-,+] => [2,1] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[+,-] => [1,2] => [2] => ([],2)
=> ? = 0 + 1
[-,-] => [1,2] => [2] => ([],2)
=> ? = 0 + 1
[+,+,+] => [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[-,+,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[+,-,+] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[-,-,+] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[-,+,-] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[+,-,-] => [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[-,-,-] => [1,2,3] => [3] => ([],3)
=> ? = 0 + 1
[+,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[-,3,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[2,1,+] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[3,+,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[3,-,1] => [3,1,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
[+,+,+,+] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[-,+,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,-,+,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,+,-,+] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[-,-,+,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,+,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,+,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,-,-,+] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,-,+,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,+,-,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[-,-,-,+] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[-,-,+,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[-,+,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[+,-,-,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[-,-,-,-] => [1,2,3,4] => [4] => ([],4)
=> ? = 0 + 1
[+,+,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,+,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,-,4,3] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,-,4,3] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[+,3,2,+] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,3,2,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,3,2,-] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,3,2,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[+,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,3,4,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[+,4,2,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,4,2,3] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,4,+,2] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,4,+,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[+,4,-,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[-,4,-,2] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[2,1,+,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,-,+] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,1,+,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,1,-,-] => [2,1,3,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[2,1,4,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[2,3,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,3,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[2,4,+,1] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[2,4,-,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[3,1,2,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,1,2,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[3,+,1,+] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,-,1,+] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[3,+,1,-] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[3,-,1,-] => [3,1,2,4] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[3,+,4,1] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
[4,1,+,2] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,+,1,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[4,+,+,1] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,+,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,-,+,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,-,+,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,+,-,+] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,+,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,4,3,+] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,+,5,+,3] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,3,2,+,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,4,2,3,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,4,+,2,+] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,5,2,3,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,5,2,+,3] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,5,+,2,4] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[+,5,+,+,2] => [1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,+,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,1,2,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,+,1,+,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,2,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,+,2,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,+,1,3,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,+,+,1,+] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,2,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,+,2,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,1,+,+,2] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,+,1,3,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,+,1,+,3] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,+,+,1,4] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[5,+,+,+,1] => [2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[-,+,+,+,+,+] => [2,3,4,5,6,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[+,-,+,+,+,+] => [1,3,4,5,6,2] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000181
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 3% ●values known / values provided: 3%●distinct values known / distinct values provided: 33%
Values
[+,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[-,+] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[+,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[-,-] => [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[+,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,+,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,-,+] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,+,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[-,-,-] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[+,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[-,3,2] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,+] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,-] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,+,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3,-,1] => [3,2,1] => [3,1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[+,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[-,+,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[-,-,4,3] => [1,2,4,3] => [1,2,4,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,3,2,+] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[-,3,2,+] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[+,3,2,-] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[-,3,2,-] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[+,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[-,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[+,4,2,3] => [1,4,2,3] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[-,4,2,3] => [1,4,2,3] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[+,4,+,2] => [1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[-,4,+,2] => [1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[+,4,-,2] => [1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[-,4,-,2] => [1,4,3,2] => [1,4,2,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[2,1,+,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,-,+] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,+,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
[2,3,1,+] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,3,1,-] => [2,3,1,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[2,4,+,1] => [2,4,3,1] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[2,4,-,1] => [2,4,3,1] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[3,1,2,+] => [3,1,2,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[3,1,2,-] => [3,1,2,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[3,+,1,+] => [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[3,-,1,+] => [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[3,+,1,-] => [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[3,-,1,-] => [3,2,1,4] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 0 + 1
[3,+,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,-,4,1] => [3,2,4,1] => [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,1,2] => [3,4,1,2] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[4,1,+,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,1,-,2] => [4,1,3,2] => [3,4,2,1] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,+,1,3] => [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[4,-,1,3] => [4,2,1,3] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 0 + 1
[4,+,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,-,+,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,+,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,-,-,1] => [4,2,3,1] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
[+,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,-,+,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-,+,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,-,-,+,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,-,+,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,-,+,+,-] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,-,-,+] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[+,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-,+,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[+,-,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[+,+,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-,-,+,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-,+,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[+,-,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[-,-,-,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
[+,+,4,3,+] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 0 + 1
Description
The number of connected components of the Hasse diagram for the poset.
The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001890The maximum magnitude of the Möbius function of a poset. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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