searching the database
Your data matches 73 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000257
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000257: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> 0
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> 0
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 0
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> 0
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 0
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 0
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 0
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 0
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 0
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 0
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> 0
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> 0
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 0
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 0
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 0
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1]
=> 0
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> []
=> 0
Description
The number of distinct parts of a partition that occur at least twice.
See Section 3.3.1 of [2].
Matching statistic: St001722
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 33%
Values
[2,1] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[2,2] => [2] => [1,1,0,0]
=> 1100 => 1 = 0 + 1
[3,1] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 0 + 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1 + 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[2,3] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[3,2] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[4,1] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? = 0 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? = 0 + 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 0 + 1
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 0 + 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 + 1
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 0 + 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 1 + 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 1 + 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1 + 1
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 0 + 1
[2,2,2] => [3] => [1,1,1,0,0,0]
=> 111000 => 1 = 0 + 1
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,4] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 1 + 1
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[3,3] => [2] => [1,1,0,0]
=> 1100 => 1 = 0 + 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[4,2] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[5,1] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 111100001010 => ? = 0 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 111000101100 => ? = 0 + 1
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => ? = 0 + 1
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => ? = 0 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 110010111000 => ? = 1 + 1
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => ? = 0 + 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => ? = 0 + 1
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 0 + 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => ? = 0 + 1
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 0 + 1
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 0 + 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 101011110000 => ? = 1 + 1
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => ? = 1 + 1
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 1 + 1
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => ? = 0 + 1
[1,2,2,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 0 + 1
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 1 + 1
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 0 + 1
[1,3,3] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[1,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 101111100000 => ? = 1 + 1
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => ? = 1 + 1
[2,1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => ? = 1 + 1
[2,1,1,3] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 10110010 => ? = 1 + 1
[2,1,2,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 1 + 1
[2,1,2,2] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 1 + 1
[2,1,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 10101010 => ? = 1 + 1
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,2,1,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => ? = 1 + 1
[2,2,1,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 11001010 => ? = 0 + 1
[2,2,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 0 + 1
[2,2,3] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[2,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 0 + 1
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,5] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[3,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 1 + 1
[3,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[3,2,2] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[3,3,1] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
[3,4] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[4,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[4,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[4,3] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[5,1,1] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[5,2] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[6,1] => [1,1] => [1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,4,3] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[2,3,3] => [1,2] => [1,0,1,1,0,0]
=> 101100 => 1 = 0 + 1
[2,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[3,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[3,3,2] => [2,1] => [1,1,0,0,1,0]
=> 110010 => 1 = 0 + 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence $01$, or a trailing $0$. A peak is a subsequence $10$ or a trailing $1$. Let $P$ be the lattice on binary words of length $n$, where the covering elements of a word are obtained by replacing a valley with a peak. An interval $[w_1, w_2]$ in $P$ is small if $w_2$ is obtained from $w_1$ by replacing some valleys with peaks.
This statistic counts the number of chains $w = w_1 < \dots < w_d = 1\dots 1$ to the top element of minimal length.
For example, there are two such chains for the word $0110$:
$$ 0110 < 1011 < 1101 < 1110 < 1111 $$
and
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St000205
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[2,1] => ([(0,2),(1,2)],3)
=> [2]
=> 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> [2]
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 0
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 0
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> ? = 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
[2,3] => ([(2,4),(3,4)],5)
=> [2]
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 0
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 0
[1,1,2,1,1] => ([(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)
=> [12]
=> ? = 0
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> 0
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> 0
[1,2,1,1,1] => ([(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)
=> [13]
=> ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> 0
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> ? = 0
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> 0
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> ? = 0
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 0
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 0
[2,1,1,1,1] => ([(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)
=> [14]
=> ? = 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> ? = 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> ? = 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 0
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 0
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 0
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> 0
[2,4] => ([(3,5),(4,5)],6)
=> [2]
=> 0
[3,1,1,1] => ([(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)
=> [12]
=> ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> 0
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [3]
=> 0
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> ? = 0
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 0
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> 0
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> ? = 0
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> ? = 0
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> ? = 0
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 0
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 0
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> ? = 0
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> 0
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> ? = 0
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> 0
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 0
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> ? = 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> 0
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> ? = 0
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 0
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> ? = 0
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4]
=> 0
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> ? = 0
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> ? = 0
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> 0
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 0
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> 0
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> 0
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> ? = 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> ? = 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> ? = 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> ? = 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> 0
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> ? = 0
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 0
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> 0
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> ? = 0
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> 0
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> 0
[2,5] => ([(4,6),(5,6)],7)
=> [2]
=> 0
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ? = 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> ? = 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> 0
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> ? = 0
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> 0
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 0
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> [3]
=> 0
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> ? = 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 0
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> ? = 0
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> [4]
=> 0
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [5]
=> 0
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[2,1] => ([(0,2),(1,2)],3)
=> [2]
=> 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> [2]
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> 0
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> 0
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> ? = 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 0
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> 0
[2,3] => ([(2,4),(3,4)],5)
=> [2]
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> 0
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 0
[1,1,2,1,1] => ([(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)
=> [12]
=> ? = 0
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> 0
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> 0
[1,2,1,1,1] => ([(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)
=> [13]
=> ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> 0
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> ? = 0
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> 0
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> ? = 0
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 0
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 0
[2,1,1,1,1] => ([(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)
=> [14]
=> ? = 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> ? = 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> ? = 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> 0
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> ? = 0
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 0
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> 0
[2,4] => ([(3,5),(4,5)],6)
=> [2]
=> 0
[3,1,1,1] => ([(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)
=> [12]
=> ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> 0
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [3]
=> 0
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> ? = 0
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> 0
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> 0
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> ? = 0
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> ? = 0
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> ? = 0
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 0
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 0
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> ? = 0
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> 0
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> ? = 0
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> 0
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 0
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> ? = 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> 0
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> ? = 0
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 0
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> ? = 0
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4]
=> 0
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> ? = 0
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> ? = 0
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> 0
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 0
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> 0
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> 0
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> ? = 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> ? = 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> ? = 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> ? = 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> 0
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> ? = 0
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 0
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> 0
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> ? = 0
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> 0
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> 0
[2,5] => ([(4,6),(5,6)],7)
=> [2]
=> 0
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> ? = 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> ? = 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> ? = 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> 0
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> ? = 0
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> 0
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 0
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> [3]
=> 0
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> ? = 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> ? = 0
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> ? = 0
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> [4]
=> 0
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [5]
=> 0
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St000644
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000644: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000644: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[2,1] => ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 0
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 0
[2,2] => ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 0
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 0
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 0
[2,3] => ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 0
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,2,1,1] => ([(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)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[1,2,1,1,1] => ([(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)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0
[2,1,1,1,1] => ([(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)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 0
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[2,4] => ([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 0
[3,1,1,1] => ([(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)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 0
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 0
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 0
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4]
=> [1,1,1,1]
=> 0
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 0
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 0
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 0
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 0
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[2,5] => ([(4,6),(5,6)],7)
=> [2]
=> [1,1]
=> 0
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 0
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 0
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> 0
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> [4]
=> [1,1,1,1]
=> 0
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 0
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 0
Description
The number of graphs with given frequency partition.
The frequency partition of a graph on $n$ vertices is the partition obtained from its degree sequence by recording and sorting the frequencies of the numbers that occur.
For example, the complete graph on $n$ vertices has frequency partition $(n)$. The path on $n$ vertices has frequency partition $(n-2,2)$, because its degree sequence is $(2,\dots,2,1,1)$. The star graph on $n$ vertices has frequency partition is $(n-1, 1)$, because its degree sequence is $(n-1,1,\dots,1)$.
There are two graphs having frequency partition $(2,1)$: the path and an edge together with an isolated vertex.
Matching statistic: St000207
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[2,1] => ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2] => ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,3] => ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,1,1] => ([(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)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,1,1,1] => ([(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)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1] => ([(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)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,4] => ([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,1,1] => ([(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)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,5] => ([(4,6),(5,6)],7)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[2,1] => ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2] => ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,3] => ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,1,1] => ([(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)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,1,1,1] => ([(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)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1] => ([(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)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,4] => ([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,1,1] => ([(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)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,5] => ([(4,6),(5,6)],7)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St001711
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001711: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Mp00275: Graphs —to edge-partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001711: Integer partitions ⟶ ℤResult quality: 24% ●values known / values provided: 24%●distinct values known / distinct values provided: 33%
Values
[2,1] => ([(0,2),(1,2)],3)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2] => ([(1,3),(2,3)],4)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,3] => ([(2,4),(3,4)],5)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,1,1] => ([(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)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,1,1,1] => ([(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)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1] => ([(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)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,4] => ([(3,5),(4,5)],6)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,1,1] => ([(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)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,3] => ([(2,5),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [19]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,3,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [20]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [16]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [17]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [11]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
[2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[2,5] => ([(4,6),(5,6)],7)
=> [2]
=> [1,1]
=> 1 = 0 + 1
[3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [18]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [7]
=> [1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [8]
=> [1,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[3,4] => ([(3,6),(4,6),(5,6)],7)
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [15]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1 + 1
[4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> ? = 0 + 1
[4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [5]
=> [1,1,1,1,1]
=> 1 = 0 + 1
[6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> [6]
=> [1,1,1,1,1,1]
=> 1 = 0 + 1
Description
The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation.
Let $\alpha$ be any permutation of cycle type $\lambda$. This statistic is the number of permutations $\pi$ such that
$$ \alpha\pi\alpha^{-1} = \pi^2.$$
Matching statistic: St000455
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Mp00172: Integer compositions —rotate back to front⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 33%
Values
[2,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,1] => [1,1,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[2,1,1] => [1,2,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[2,2] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[3,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,1,2,1] => [1,1,1,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,2,1,1] => [1,1,2,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,2,2] => [2,1,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[1,3,1] => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,1,1,1] => [1,2,1,1] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1
[2,1,2] => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0
[2,2,1] => [1,2,2] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,3] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[3,1,1] => [1,3,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[3,2] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[4,1] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 0
[1,1,1,2,1] => [1,1,1,1,2] => [2,1,1,1,1] => ([(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
[1,1,2,1,1] => [1,1,1,2,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,1,2,2] => [2,1,1,2] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,1,3,1] => [1,1,1,3] => [3,1,1,1] => ([(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
[1,2,1,1,1] => [1,1,2,1,1] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[1,2,1,2] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,2,2,1] => [1,1,2,2] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,2,3] => [3,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,3,1,1] => [1,1,3,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,3,2] => [2,1,3] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[1,4,1] => [1,1,4] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,1,1,1,1] => [1,2,1,1,1] => [1,1,2,1,1] => ([(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)
=> ? = 1
[2,1,1,2] => [2,2,1,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,1,2,1] => [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[2,1,3] => [3,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[2,2,1,1] => [1,2,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[2,2,2] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[2,3,1] => [1,2,3] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,4] => [4,2] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[3,1,1,1] => [1,3,1,1] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1
[3,1,2] => [2,3,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0
[3,2,1] => [1,3,2] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[3,3] => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[4,1,1] => [1,4,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[4,2] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[5,1] => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[1,1,1,1,2,1] => [1,1,1,1,1,2] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,1,2,1,1] => [1,1,1,1,2,1] => [1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,1,2,2] => [2,1,1,1,2] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,1,3,1] => [1,1,1,1,3] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,2,1,1,1] => [1,1,1,2,1,1] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,1,2,1,2] => [2,1,1,2,1] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,2,2,1] => [1,1,1,2,2] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,2,3] => [3,1,1,2] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,3,1,1] => [1,1,1,3,1] => [1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,1,3,2] => [2,1,1,3] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,1,4,1] => [1,1,1,4] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,2,1,1,1,1] => [1,1,2,1,1,1] => [1,1,1,2,1,1] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,1,1,2] => [2,1,2,1,1] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,1,2,1] => [1,1,2,1,2] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,2,1,3] => [3,1,2,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,2,2,1,1] => [1,1,2,2,1] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,2,2,2] => [2,1,2,2] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,2,3,1] => [1,1,2,3] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,2,4] => [4,1,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,3,1,1,1] => [1,1,3,1,1] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[1,3,1,2] => [2,1,3,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,3,2,1] => [1,1,3,2] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,3,3] => [3,1,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,4,1,1] => [1,1,4,1] => [1,1,1,4] => ([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[1,4,2] => [2,1,4] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[1,5,1] => [1,1,5] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[2,1,1,1,1,1] => [1,2,1,1,1,1] => [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,1,1,2] => [2,2,1,1,1] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,1,2,1] => [1,2,1,1,2] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,1,3] => [3,2,1,1] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,2,1,1] => [1,2,1,2,1] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,2,2] => [2,2,1,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,3,1] => [1,2,1,3] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,1,4] => [4,2,1] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,2,1,1,1] => [1,2,2,1,1] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[2,2,1,2] => [2,2,2,1] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,2,2,1] => [1,2,2,2] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,2,3] => [3,2,2] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,3,1,1] => [1,2,3,1] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,3,2] => [2,2,3] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[2,4,1] => [1,2,4] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[2,5] => [5,2] => [2,5] => ([(4,6),(5,6)],7)
=> 0
[3,1,1,1,1] => [1,3,1,1,1] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,1,1,2] => [2,3,1,1] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,1,2,1] => [1,3,1,2] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 1
[3,1,3] => [3,3,1] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 0
[3,3,1] => [1,3,3] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[3,4] => [4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
=> 0
[4,2,1] => [1,4,2] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 0
[4,3] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[5,1,1] => [1,5,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
=> 0
[5,2] => [2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
[6,1] => [1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000782
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Values
[2,1] => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[2,2] => [2] => [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 0 + 1
[3,1] => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 0 + 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 0 + 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[2,3] => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[3,2] => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[4,1] => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> ? = 0 + 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> ? = 0 + 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 1
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 0 + 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 0 + 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> ? = 0 + 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 0 + 1
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1 + 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> ? = 1 + 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 1 + 1
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 0 + 1
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1 = 0 + 1
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,4] => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[3,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1 + 1
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[3,3] => [2] => [1,1,0,0]
=> [(1,4),(2,3)]
=> ? = 0 + 1
[4,1,1] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[4,2] => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[5,1] => [1,1] => [1,0,1,0]
=> [(1,2),(3,4)]
=> ? = 0 + 1
[1,1,1,1,2,1] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0 + 1
[1,1,1,2,1,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,12),(10,11)]
=> ? = 0 + 1
[1,1,1,2,2] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> ? = 0 + 1
[1,1,1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> ? = 0 + 1
[1,1,2,1,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,6),(7,12),(8,11),(9,10)]
=> ? = 1 + 1
[1,1,2,1,2] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> ? = 0 + 1
[1,1,2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> ? = 0 + 1
[1,1,2,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 0 + 1
[1,1,3,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> ? = 0 + 1
[1,1,3,2] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 0 + 1
[1,1,4,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> ? = 0 + 1
[1,2,1,1,1,1] => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> ? = 1 + 1
[1,2,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> ? = 1 + 1
[1,2,1,2,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> ? = 1 + 1
[1,2,1,3] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 0 + 1
[1,2,2,1,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> ? = 0 + 1
[1,2,2,2] => [1,3] => [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 0 + 1
[1,2,3,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 0 + 1
[1,2,4] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[1,3,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[1,3,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 0 + 1
[1,3,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> ? = 0 + 1
[1,3,3] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[1,4,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 0 + 1
[1,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[1,5,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,1,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 1 + 1
[2,1,1,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> ? = 1 + 1
[2,1,4] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,2,3] => [2,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[2,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[3,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[3,2,2] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[3,3,1] => [2,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[4,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[4,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[5,1,1] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[1,4,3] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[2,3,3] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[2,4,2] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[3,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[3,3,2] => [2,1] => [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 0 + 1
[3,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[4,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[4,2,2] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[4,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 0 + 1
[6,1,1] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[7,1,1] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
[8,1,1] => [1,2] => [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 0 + 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000552The number of cut vertices of a graph. St001793The difference between the clique number and the chromatic number of a graph. St000553The number of blocks of a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000916The packing number of a graph. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St000258The burning number of a graph. St000918The 2-limited packing number of a graph. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000379The number of Hamiltonian cycles in a graph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001797The number of overfull subgraphs of a graph. St000544The cop number of a graph. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001367The smallest number which does not occur as degree of a vertex in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001496The number of graphs with the same Laplacian spectrum as the given graph. St000264The girth of a graph, which is not a tree. St000322The skewness of a graph. St001307The number of induced stars on four vertices in a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000315The number of isolated vertices of a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001765The number of connected components of the friends and strangers graph. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001871The number of triconnected components of a graph. St000286The number of connected components of the complement of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000031The number of cycles in the cycle decomposition of a permutation. St000741The Colin de Verdière graph invariant.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!