Identifier
Values
[1,1] => [[1,1],[]] => ([(0,1)],2) => 1
[2] => [[2],[]] => ([(0,1)],2) => 1
[1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 1
[1,2] => [[2,1],[]] => ([(0,1),(0,2)],3) => 2
[2,1] => [[2,2],[1]] => ([(0,2),(1,2)],3) => 2
[3] => [[3],[]] => ([(0,2),(2,1)],3) => 1
[1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,2] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 3
[1,2,1] => [[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4) => 5
[1,3] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 3
[2,1,1] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => 3
[2,2] => [[3,2],[1]] => ([(0,3),(1,2),(1,3)],4) => 5
[3,1] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => 3
[4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 1
[1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,2] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 4
[1,1,2,1] => [[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 9
[1,1,3] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 6
[1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 9
[1,2,2] => [[3,2,1],[1]] => ([(0,3),(0,4),(1,2),(1,4)],5) => 16
[1,3,1] => [[3,3,1],[2]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 11
[1,4] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 4
[2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[2,1,2] => [[3,2,2],[1,1]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 11
[2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 16
[2,3] => [[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 9
[3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 6
[3,2] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 9
[4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 4
[5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,1,1,2] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 5
[1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => 14
[1,1,1,3] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 10
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 19
[1,1,2,2] => [[3,2,1,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 35
[1,1,3,1] => [[3,3,1,1],[2]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6) => 26
[1,1,4] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 10
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 14
[1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 40
[1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 61
[1,2,3] => [[4,2,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 35
[1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 26
[1,3,2] => [[4,3,1],[2]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 40
[1,4,1] => [[4,4,1],[3]] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6) => 19
[1,5] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 5
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 5
[2,1,1,2] => [[3,2,2,2],[1,1,1]] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6) => 19
[2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 40
[2,1,3] => [[4,2,2],[1,1]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6) => 26
[2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6) => 35
[2,2,2] => [[4,3,2],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 61
[2,3,1] => [[4,4,2],[3,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 40
[2,4] => [[5,2],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => 14
[3,1,1,1] => [[3,3,3,3],[2,2,2]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 10
[3,1,2] => [[4,3,3],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 26
[3,2,1] => [[4,4,3],[3,2]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6) => 35
[3,3] => [[5,3],[2]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 19
[4,1,1] => [[4,4,4],[3,3]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 10
[4,2] => [[5,4],[3]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 14
[5,1] => [[5,5],[4]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 5
[6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[1,1,1,1,1,2] => [[2,1,1,1,1,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => 6
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7) => 20
[1,1,1,1,3] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 15
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7) => 34
[1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) => 64
[1,1,1,3,1] => [[3,3,1,1,1],[2]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7) => 50
[1,1,1,4] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7) => 20
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7) => 34
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7) => 99
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7) => 155
[1,1,2,3] => [[4,2,1,1],[1]] => ([(0,5),(0,6),(1,4),(1,6),(4,2),(5,3)],7) => 90
[1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7) => 71
[1,1,3,2] => [[4,3,1,1],[2]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) => 111
[1,1,4,1] => [[4,4,1,1],[3]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) => 55
[1,1,5] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 15
[1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7) => 20
[1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7) => 78
[1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7) => 169
[1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) => 111
[1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7) => 155
[1,2,2,2] => [[4,3,2,1],[2,1]] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5)],7) => 272
[1,2,3,1] => [[4,4,2,1],[3,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7) => 181
[1,2,4] => [[5,2,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) => 64
[1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7) => 50
[1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,3),(0,5),(1,2),(1,4),(4,6),(5,6)],7) => 132
[1,3,2,1] => [[4,4,3,1],[3,2]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7) => 181
[1,3,3] => [[5,3,1],[2]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7) => 99
[1,4,1,1] => [[4,4,4,1],[3,3]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) => 55
[1,4,2] => [[5,4,1],[3]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7) => 78
[1,5,1] => [[5,5,1],[4]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7) => 29
[1,6] => [[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => 6
[2,1,1,1,1,1] => [[2,2,2,2,2,2],[1,1,1,1,1]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => 6
[2,1,1,1,2] => [[3,2,2,2,2],[1,1,1,1]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7) => 29
[2,1,1,2,1] => [[3,3,2,2,2],[2,1,1,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7) => 78
[2,1,1,3] => [[4,2,2,2],[1,1,1]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) => 55
[2,1,2,1,1] => [[3,3,3,2,2],[2,2,1,1]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7) => 99
[2,1,2,2] => [[4,3,2,2],[2,1,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7) => 181
[2,1,3,1] => [[4,4,2,2],[3,1,1]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 132
>>> Load all 153 entries. <<<
[2,1,4] => [[5,2,2],[1,1]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7) => 50
[2,2,1,1,1] => [[3,3,3,3,2],[2,2,2,1]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7) => 64
[2,2,1,2] => [[4,3,3,2],[2,2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7) => 181
[2,2,2,1] => [[4,4,3,2],[3,2,1]] => ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7) => 272
[2,2,3] => [[5,3,2],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7) => 155
[2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) => 111
[2,3,2] => [[5,4,2],[3,1]] => ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7) => 169
[2,4,1] => [[5,5,2],[4,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7) => 78
[2,5] => [[6,2],[1]] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7) => 20
[3,1,1,1,1] => [[3,3,3,3,3],[2,2,2,2]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => 15
[3,1,1,2] => [[4,3,3,3],[2,2,2]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) => 55
[3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) => 111
[3,1,3] => [[5,3,3],[2,2]] => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7) => 71
[3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7) => 90
[3,2,2] => [[5,4,3],[3,2]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7) => 155
[3,3,1] => [[5,5,3],[4,2]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7) => 99
[3,4] => [[6,3],[2]] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7) => 34
[4,1,1,1] => [[4,4,4,4],[3,3,3]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => 20
[4,1,2] => [[5,4,4],[3,3]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7) => 50
[4,2,1] => [[5,5,4],[4,3]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7) => 64
[4,3] => [[6,4],[3]] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7) => 34
[5,1,1] => [[5,5,5],[4,4]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => 15
[5,2] => [[6,5],[4]] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7) => 20
[6,1] => [[6,6],[5]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => 6
[7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 1
[1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 1
[1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => 7
[1,1,1,1,1,3] => [[3,1,1,1,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8) => 21
[1,1,1,1,4] => [[4,1,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8) => 35
[1,1,1,5] => [[5,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8) => 35
[1,1,6] => [[6,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8) => 21
[1,7] => [[7,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => 7
[8] => [[8],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 1
[1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 1
[1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => 8
[1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) => 28
[1,1,1,1,1,4] => [[4,1,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9) => 56
[1,1,1,1,5] => [[5,1,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9) => 70
[1,1,1,6] => [[6,1,1,1],[]] => ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9) => 56
[1,1,7] => [[7,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) => 28
[1,8] => [[8,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => 8
[9] => [[9],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 1
[1,1,1,1,1,1,1,1,1,1] => [[1,1,1,1,1,1,1,1,1,1],[]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1
[1,1,1,1,1,1,1,1,2] => [[2,1,1,1,1,1,1,1,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 9
[1,1,1,1,1,1,1,3] => [[3,1,1,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10) => 36
[1,1,1,1,1,1,4] => [[4,1,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10) => 84
[1,1,1,1,1,5] => [[5,1,1,1,1,1],[]] => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10) => 126
[1,1,1,1,6] => [[6,1,1,1,1],[]] => ([(0,8),(0,9),(3,7),(4,3),(5,6),(6,1),(7,2),(8,4),(9,5)],10) => 126
[1,1,1,7] => [[7,1,1,1],[]] => ([(0,8),(0,9),(3,4),(4,6),(5,3),(6,2),(7,1),(8,7),(9,5)],10) => 84
[1,1,8] => [[8,1,1],[]] => ([(0,8),(0,9),(3,5),(4,3),(5,7),(6,4),(7,2),(8,6),(9,1)],10) => 36
[1,9] => [[9,1],[]] => ([(0,2),(0,9),(3,4),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 9
[10] => [[10],[]] => ([(0,9),(2,4),(3,2),(4,6),(5,3),(6,8),(7,5),(8,1),(9,7)],10) => 1
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Description
The number of linear extensions of a poset.
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to ribbon
Description
The ribbon shape corresponding to an integer composition.
For an integer composition $(a_1, \dots, a_n)$, this is the ribbon shape whose $i$th row from the bottom has $a_i$ cells.
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cell poset
Description
The Young diagram of a skew partition regarded as a poset.
This is the poset on the cells of the Young diagram, such that a cell $d$ is greater than a cell $c$ if the entry in $d$ must be larger than the entry of $c$ in any standard Young tableau on the skew partition.