Identifier
Values
0 => [2] => [[2],[]] => ([(0,1)],2) => 3
1 => [1,1] => [[1,1],[]] => ([(0,1)],2) => 3
00 => [3] => [[3],[]] => ([(0,2),(2,1)],3) => 4
01 => [2,1] => [[2,2],[1]] => ([(0,2),(1,2)],3) => 5
10 => [1,2] => [[2,1],[]] => ([(0,1),(0,2)],3) => 5
11 => [1,1,1] => [[1,1,1],[]] => ([(0,2),(2,1)],3) => 4
000 => [4] => [[4],[]] => ([(0,3),(2,1),(3,2)],4) => 5
001 => [3,1] => [[3,3],[2]] => ([(0,3),(1,2),(2,3)],4) => 7
010 => [2,2] => [[3,2],[1]] => ([(0,3),(1,2),(1,3)],4) => 8
011 => [2,1,1] => [[2,2,2],[1,1]] => ([(0,3),(1,2),(2,3)],4) => 7
100 => [1,3] => [[3,1],[]] => ([(0,2),(0,3),(3,1)],4) => 7
101 => [1,2,1] => [[2,2,1],[1]] => ([(0,3),(1,2),(1,3)],4) => 8
110 => [1,1,2] => [[2,1,1],[]] => ([(0,2),(0,3),(3,1)],4) => 7
111 => [1,1,1,1] => [[1,1,1,1],[]] => ([(0,3),(2,1),(3,2)],4) => 5
0000 => [5] => [[5],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 6
0001 => [4,1] => [[4,4],[3]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 9
0010 => [3,2] => [[4,3],[2]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 11
0011 => [3,1,1] => [[3,3,3],[2,2]] => ([(0,3),(1,2),(2,4),(3,4)],5) => 10
0100 => [2,3] => [[4,2],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 11
0101 => [2,2,1] => [[3,3,2],[2,1]] => ([(0,4),(1,3),(2,3),(2,4)],5) => 13
0110 => [2,1,2] => [[3,2,2],[1,1]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 12
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => ([(0,4),(1,2),(2,3),(3,4)],5) => 9
1000 => [1,4] => [[4,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 9
1001 => [1,3,1] => [[3,3,1],[2]] => ([(0,4),(1,2),(1,3),(3,4)],5) => 12
1010 => [1,2,2] => [[3,2,1],[1]] => ([(0,3),(0,4),(1,2),(1,4)],5) => 13
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]] => ([(0,3),(1,2),(1,4),(3,4)],5) => 11
1100 => [1,1,3] => [[3,1,1],[]] => ([(0,3),(0,4),(3,2),(4,1)],5) => 10
1101 => [1,1,2,1] => [[2,2,1,1],[1]] => ([(0,4),(1,2),(1,4),(2,3)],5) => 11
1110 => [1,1,1,2] => [[2,1,1,1],[]] => ([(0,2),(0,4),(3,1),(4,3)],5) => 9
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]] => ([(0,4),(2,3),(3,1),(4,2)],5) => 6
00000 => [6] => [[6],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 7
00001 => [5,1] => [[5,5],[4]] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6) => 11
00010 => [4,2] => [[5,4],[3]] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6) => 14
00011 => [4,1,1] => [[4,4,4],[3,3]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 13
00100 => [3,3] => [[5,3],[2]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 15
00101 => [3,2,1] => [[4,4,3],[3,2]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6) => 18
00110 => [3,1,2] => [[4,3,3],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 17
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6) => 13
01000 => [2,4] => [[5,2],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => 14
01001 => [2,3,1] => [[4,4,2],[3,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 19
01010 => [2,2,2] => [[4,3,2],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 21
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6) => 18
01100 => [2,1,3] => [[4,2,2],[1,1]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6) => 17
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 19
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6) => 16
01111 => [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) => 11
10000 => [1,5] => [[5,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 11
10001 => [1,4,1] => [[4,4,1],[3]] => ([(0,5),(1,2),(1,4),(3,5),(4,3)],6) => 16
10010 => [1,3,2] => [[4,3,1],[2]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 19
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]] => ([(0,4),(1,2),(1,3),(3,5),(4,5)],6) => 17
10100 => [1,2,3] => [[4,2,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 18
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6) => 21
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]] => ([(0,4),(0,5),(1,2),(1,3),(3,5)],6) => 19
10111 => [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
11000 => [1,1,4] => [[4,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 13
11001 => [1,1,3,1] => [[3,3,1,1],[2]] => ([(0,5),(1,3),(1,4),(3,5),(4,2)],6) => 17
11010 => [1,1,2,2] => [[3,2,1,1],[1]] => ([(0,3),(0,5),(1,4),(1,5),(4,2)],6) => 18
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6) => 15
11100 => [1,1,1,3] => [[3,1,1,1],[]] => ([(0,4),(0,5),(3,2),(4,3),(5,1)],6) => 13
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]] => ([(0,5),(1,4),(1,5),(3,2),(4,3)],6) => 14
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6) => 11
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 7
000000 => [7] => [[7],[]] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 8
000001 => [6,1] => [[6,6],[5]] => ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7) => 13
000010 => [5,2] => [[6,5],[4]] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7) => 17
000011 => [5,1,1] => [[5,5,5],[4,4]] => ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7) => 16
000100 => [4,3] => [[6,4],[3]] => ([(0,4),(1,5),(1,6),(3,6),(4,3),(5,2)],7) => 19
000101 => [4,2,1] => [[5,5,4],[4,3]] => ([(0,5),(1,5),(1,6),(2,3),(3,4),(4,6)],7) => 23
000110 => [4,1,2] => [[5,4,4],[3,3]] => ([(0,5),(1,2),(1,4),(3,6),(4,6),(5,3)],7) => 22
000111 => [4,1,1,1] => [[4,4,4,4],[3,3,3]] => ([(0,5),(1,4),(2,6),(3,6),(4,2),(5,3)],7) => 17
001000 => [3,4] => [[6,3],[2]] => ([(0,3),(1,5),(1,6),(3,6),(4,2),(5,4)],7) => 19
001001 => [3,3,1] => [[5,5,3],[4,2]] => ([(0,6),(1,3),(2,4),(2,5),(3,5),(4,6)],7) => 26
001010 => [3,2,2] => [[5,4,3],[3,2]] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7) => 29
001011 => [3,2,1,1] => [[4,4,4,3],[3,3,2]] => ([(0,5),(0,6),(1,4),(2,3),(3,5),(4,6)],7) => 25
001100 => [3,1,3] => [[5,3,3],[2,2]] => ([(0,4),(1,3),(1,5),(3,6),(4,6),(5,2)],7) => 24
001101 => [3,1,2,1] => [[4,4,3,3],[3,2,2]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) => 27
001110 => [3,1,1,2] => [[4,3,3,3],[2,2,2]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) => 23
001111 => [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) => 16
010000 => [2,5] => [[6,2],[1]] => ([(0,6),(1,5),(1,6),(3,4),(4,2),(5,3)],7) => 17
010001 => [2,4,1] => [[5,5,2],[4,1]] => ([(0,5),(1,6),(2,3),(2,5),(3,4),(4,6)],7) => 25
010010 => [2,3,2] => [[5,4,2],[3,1]] => ([(0,5),(1,3),(1,6),(2,4),(2,5),(4,6)],7) => 30
010011 => [2,3,1,1] => [[4,4,4,2],[3,3,1]] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6)],7) => 27
010100 => [2,2,3] => [[5,3,2],[2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,6),(3,4)],7) => 29
010101 => [2,2,2,1] => [[4,4,3,2],[3,2,1]] => ([(0,5),(1,4),(2,4),(2,6),(3,5),(3,6)],7) => 34
010110 => [2,2,1,2] => [[4,3,3,2],[2,2,1]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7) => 31
010111 => [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) => 23
011000 => [2,1,4] => [[5,2,2],[1,1]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7) => 22
011001 => [2,1,3,1] => [[4,4,2,2],[3,1,1]] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7) => 29
011010 => [2,1,2,2] => [[4,3,2,2],[2,1,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7) => 31
011011 => [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) => 26
011100 => [2,1,1,3] => [[4,2,2,2],[1,1,1]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) => 23
011101 => [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) => 25
011110 => [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) => 20
011111 => [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) => 13
100000 => [1,6] => [[6,1],[]] => ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7) => 13
100001 => [1,5,1] => [[5,5,1],[4]] => ([(0,6),(1,2),(1,5),(3,6),(4,3),(5,4)],7) => 20
100010 => [1,4,2] => [[5,4,1],[3]] => ([(0,4),(0,6),(1,2),(1,5),(3,6),(5,3)],7) => 25
100011 => [1,4,1,1] => [[4,4,4,1],[3,3]] => ([(0,4),(1,2),(1,5),(3,6),(4,6),(5,3)],7) => 23
100100 => [1,3,3] => [[5,3,1],[2]] => ([(0,5),(0,6),(1,3),(1,4),(4,6),(5,2)],7) => 26
100101 => [1,3,2,1] => [[4,4,3,1],[3,2]] => ([(0,5),(1,5),(1,6),(2,3),(2,4),(4,6)],7) => 31
100110 => [1,3,1,2] => [[4,3,3,1],[2,2]] => ([(0,3),(0,5),(1,2),(1,4),(4,6),(5,6)],7) => 29
>>> Load all 144 entries. <<<
100111 => [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) => 22
101000 => [1,2,4] => [[5,2,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) => 23
101001 => [1,2,3,1] => [[4,4,2,1],[3,1]] => ([(0,6),(1,3),(1,5),(2,4),(2,5),(4,6)],7) => 31
101010 => [1,2,2,2] => [[4,3,2,1],[2,1]] => ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5)],7) => 34
101011 => [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) => 29
101100 => [1,2,1,3] => [[4,2,2,1],[1,1]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) => 27
101101 => [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) => 30
101110 => [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) => 25
101111 => [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) => 17
110000 => [1,1,5] => [[5,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 16
110001 => [1,1,4,1] => [[4,4,1,1],[3]] => ([(0,6),(1,4),(1,5),(3,6),(4,2),(5,3)],7) => 23
110010 => [1,1,3,2] => [[4,3,1,1],[2]] => ([(0,4),(0,6),(1,3),(1,5),(3,6),(5,2)],7) => 27
110011 => [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) => 24
110100 => [1,1,2,3] => [[4,2,1,1],[1]] => ([(0,5),(0,6),(1,4),(1,6),(4,2),(5,3)],7) => 25
110101 => [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) => 29
110110 => [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) => 26
110111 => [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) => 19
111000 => [1,1,1,4] => [[4,1,1,1],[]] => ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7) => 17
111001 => [1,1,1,3,1] => [[3,3,1,1,1],[2]] => ([(0,6),(1,3),(1,5),(3,6),(4,2),(5,4)],7) => 22
111010 => [1,1,1,2,2] => [[3,2,1,1,1],[1]] => ([(0,5),(0,6),(1,3),(1,6),(4,2),(5,4)],7) => 23
111011 => [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) => 19
111100 => [1,1,1,1,3] => [[3,1,1,1,1],[]] => ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7) => 16
111101 => [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) => 17
111110 => [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) => 13
111111 => [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) => 8
0000000 => [8] => [[8],[]] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8) => 9
1000000 => [1,7] => [[7,1],[]] => ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8) => 15
1100000 => [1,1,6] => [[6,1,1],[]] => ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8) => 19
1110000 => [1,1,1,5] => [[5,1,1,1],[]] => ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8) => 21
1111000 => [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) => 21
1111100 => [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) => 19
1111110 => [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) => 15
1111111 => [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) => 9
00000000 => [9] => [[9],[]] => ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9) => 10
10000000 => [1,8] => [[8,1],[]] => ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9) => 17
11000000 => [1,1,7] => [[7,1,1],[]] => ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9) => 22
11100000 => [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) => 25
11110000 => [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) => 26
11111000 => [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) => 25
11111100 => [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) => 22
11111110 => [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) => 17
11111111 => [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) => 10
=> [1] => [[1],[]] => ([],1) => 2
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Description
The number of antichains in a poset.
An antichain in a poset $P$ is a subset of elements of $P$ which are pairwise incomparable.
An order ideal is a subset $I$ of $P$ such that $a\in I$ and $b \leq_P a$ implies $b \in I$. Since there is a one-to-one correspondence between antichains and order ideals, this statistic is also the number of order ideals in a poset.
Map
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.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.
Map
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.