Identifier
Mp00231:
Integer compositions
—bounce path⟶
Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Images
[1] => [1,0] => [[1],[2]] => [[1],[2]]
[1,1] => [1,0,1,0] => [[1,3],[2,4]] => [[1,2],[3,4]]
[2] => [1,1,0,0] => [[1,2],[3,4]] => [[1,3],[2,4]]
[1,1,1] => [1,0,1,0,1,0] => [[1,3,5],[2,4,6]] => [[1,2,4],[3,5,6]]
[1,2] => [1,0,1,1,0,0] => [[1,3,4],[2,5,6]] => [[1,2,5],[3,4,6]]
[2,1] => [1,1,0,0,1,0] => [[1,2,5],[3,4,6]] => [[1,3,4],[2,5,6]]
[3] => [1,1,1,0,0,0] => [[1,2,3],[4,5,6]] => [[1,3,5],[2,4,6]]
[1,1,1,1] => [1,0,1,0,1,0,1,0] => [[1,3,5,7],[2,4,6,8]] => [[1,2,4,6],[3,5,7,8]]
[1,1,2] => [1,0,1,0,1,1,0,0] => [[1,3,5,6],[2,4,7,8]] => [[1,2,4,7],[3,5,6,8]]
[1,2,1] => [1,0,1,1,0,0,1,0] => [[1,3,4,7],[2,5,6,8]] => [[1,2,5,6],[3,4,7,8]]
[1,3] => [1,0,1,1,1,0,0,0] => [[1,3,4,5],[2,6,7,8]] => [[1,2,5,7],[3,4,6,8]]
[2,1,1] => [1,1,0,0,1,0,1,0] => [[1,2,5,7],[3,4,6,8]] => [[1,3,4,6],[2,5,7,8]]
[2,2] => [1,1,0,0,1,1,0,0] => [[1,2,5,6],[3,4,7,8]] => [[1,3,4,7],[2,5,6,8]]
[3,1] => [1,1,1,0,0,0,1,0] => [[1,2,3,7],[4,5,6,8]] => [[1,3,5,6],[2,4,7,8]]
[4] => [1,1,1,1,0,0,0,0] => [[1,2,3,4],[5,6,7,8]] => [[1,3,5,7],[2,4,6,8]]
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9],[2,4,6,8,10]] => [[1,2,4,6,8],[3,5,7,9,10]]
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,8],[2,4,6,9,10]] => [[1,2,4,6,9],[3,5,7,8,10]]
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => [[1,3,5,6,9],[2,4,7,8,10]] => [[1,2,4,7,8],[3,5,6,9,10]]
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => [[1,3,5,6,7],[2,4,8,9,10]] => [[1,2,4,7,9],[3,5,6,8,10]]
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,9],[2,5,6,8,10]] => [[1,2,5,6,8],[3,4,7,9,10]]
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => [[1,3,4,7,8],[2,5,6,9,10]] => [[1,2,5,6,9],[3,4,7,8,10]]
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => [[1,3,4,5,9],[2,6,7,8,10]] => [[1,2,5,7,8],[3,4,6,9,10]]
[1,4] => [1,0,1,1,1,1,0,0,0,0] => [[1,3,4,5,6],[2,7,8,9,10]] => [[1,2,5,7,9],[3,4,6,8,10]]
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => [[1,2,5,7,9],[3,4,6,8,10]] => [[1,3,4,6,8],[2,5,7,9,10]]
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => [[1,2,5,7,8],[3,4,6,9,10]] => [[1,3,4,6,9],[2,5,7,8,10]]
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => [[1,2,5,6,9],[3,4,7,8,10]] => [[1,3,4,7,8],[2,5,6,9,10]]
[2,3] => [1,1,0,0,1,1,1,0,0,0] => [[1,2,5,6,7],[3,4,8,9,10]] => [[1,3,4,7,9],[2,5,6,8,10]]
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => [[1,2,3,7,9],[4,5,6,8,10]] => [[1,3,5,6,8],[2,4,7,9,10]]
[3,2] => [1,1,1,0,0,0,1,1,0,0] => [[1,2,3,7,8],[4,5,6,9,10]] => [[1,3,5,6,9],[2,4,7,8,10]]
[4,1] => [1,1,1,1,0,0,0,0,1,0] => [[1,2,3,4,9],[5,6,7,8,10]] => [[1,3,5,7,8],[2,4,6,9,10]]
[5] => [1,1,1,1,1,0,0,0,0,0] => [[1,2,3,4,5],[6,7,8,9,10]] => [[1,3,5,7,9],[2,4,6,8,10]]
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11],[2,4,6,8,10,12]] => [[1,2,4,6,8,10],[3,5,7,9,11,12]]
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,9,10],[2,4,6,8,11,12]] => [[1,2,4,6,8,11],[3,5,7,9,10,12]]
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,8,11],[2,4,6,9,10,12]] => [[1,2,4,6,9,10],[3,5,7,8,11,12]]
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => [[1,3,5,7,8,9],[2,4,6,10,11,12]] => [[1,2,4,6,9,11],[3,5,7,8,10,12]]
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => [[1,3,5,6,9,11],[2,4,7,8,10,12]] => [[1,2,4,7,8,10],[3,5,6,9,11,12]]
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => [[1,3,5,6,9,10],[2,4,7,8,11,12]] => [[1,2,4,7,8,11],[3,5,6,9,10,12]]
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => [[1,3,5,6,7,11],[2,4,8,9,10,12]] => [[1,2,4,7,9,10],[3,5,6,8,11,12]]
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => [[1,3,5,6,7,8],[2,4,9,10,11,12]] => [[1,2,4,7,9,11],[3,5,6,8,10,12]]
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => [[1,3,4,7,9,11],[2,5,6,8,10,12]] => [[1,2,5,6,8,10],[3,4,7,9,11,12]]
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => [[1,3,4,7,9,10],[2,5,6,8,11,12]] => [[1,2,5,6,8,11],[3,4,7,9,10,12]]
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => [[1,3,4,7,8,11],[2,5,6,9,10,12]] => [[1,2,5,6,9,10],[3,4,7,8,11,12]]
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => [[1,3,4,7,8,9],[2,5,6,10,11,12]] => [[1,2,5,6,9,11],[3,4,7,8,10,12]]
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => [[1,3,4,5,9,11],[2,6,7,8,10,12]] => [[1,2,5,7,8,10],[3,4,6,9,11,12]]
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => [[1,3,4,5,9,10],[2,6,7,8,11,12]] => [[1,2,5,7,8,11],[3,4,6,9,10,12]]
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => [[1,3,4,5,6,11],[2,7,8,9,10,12]] => [[1,2,5,7,9,10],[3,4,6,8,11,12]]
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => [[1,3,4,5,6,7],[2,8,9,10,11,12]] => [[1,2,5,7,9,11],[3,4,6,8,10,12]]
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => [[1,2,5,7,9,11],[3,4,6,8,10,12]] => [[1,3,4,6,8,10],[2,5,7,9,11,12]]
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => [[1,2,5,7,9,10],[3,4,6,8,11,12]] => [[1,3,4,6,8,11],[2,5,7,9,10,12]]
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => [[1,2,5,7,8,11],[3,4,6,9,10,12]] => [[1,3,4,6,9,10],[2,5,7,8,11,12]]
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => [[1,2,5,7,8,9],[3,4,6,10,11,12]] => [[1,3,4,6,9,11],[2,5,7,8,10,12]]
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => [[1,2,5,6,9,11],[3,4,7,8,10,12]] => [[1,3,4,7,8,10],[2,5,6,9,11,12]]
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => [[1,2,5,6,9,10],[3,4,7,8,11,12]] => [[1,3,4,7,8,11],[2,5,6,9,10,12]]
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => [[1,2,5,6,7,11],[3,4,8,9,10,12]] => [[1,3,4,7,9,10],[2,5,6,8,11,12]]
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => [[1,2,5,6,7,8],[3,4,9,10,11,12]] => [[1,3,4,7,9,11],[2,5,6,8,10,12]]
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => [[1,2,3,7,9,11],[4,5,6,8,10,12]] => [[1,3,5,6,8,10],[2,4,7,9,11,12]]
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => [[1,2,3,7,9,10],[4,5,6,8,11,12]] => [[1,3,5,6,8,11],[2,4,7,9,10,12]]
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => [[1,2,3,7,8,11],[4,5,6,9,10,12]] => [[1,3,5,6,9,10],[2,4,7,8,11,12]]
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => [[1,2,3,7,8,9],[4,5,6,10,11,12]] => [[1,3,5,6,9,11],[2,4,7,8,10,12]]
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => [[1,2,3,4,9,11],[5,6,7,8,10,12]] => [[1,3,5,7,8,10],[2,4,6,9,11,12]]
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => [[1,2,3,4,9,10],[5,6,7,8,11,12]] => [[1,3,5,7,8,11],[2,4,6,9,10,12]]
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => [[1,2,3,4,5,11],[6,7,8,9,10,12]] => [[1,3,5,7,9,10],[2,4,6,8,11,12]]
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => [[1,2,3,4,5,6],[7,8,9,10,11,12]] => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14]] => [[1,2,4,6,8,10,12],[3,5,7,9,11,13,14]]
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [[1,3,5,7,9,11,12],[2,4,6,8,10,13,14]] => [[1,2,4,6,8,10,13],[3,5,7,9,11,12,14]]
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => [[1,3,5,7,9,10,13],[2,4,6,8,11,12,14]] => [[1,2,4,6,8,11,12],[3,5,7,9,10,13,14]]
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [[1,3,5,7,9,10,11],[2,4,6,8,12,13,14]] => [[1,2,4,6,8,11,13],[3,5,7,9,10,12,14]]
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [[1,3,5,7,8,11,13],[2,4,6,9,10,12,14]] => [[1,2,4,6,9,10,12],[3,5,7,8,11,13,14]]
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => [[1,3,5,7,8,11,12],[2,4,6,9,10,13,14]] => [[1,2,4,6,9,10,13],[3,5,7,8,11,12,14]]
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [[1,3,5,7,8,9,13],[2,4,6,10,11,12,14]] => [[1,2,4,6,9,11,12],[3,5,7,8,10,13,14]]
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [[1,3,5,7,8,9,10],[2,4,6,11,12,13,14]] => [[1,2,4,6,9,11,13],[3,5,7,8,10,12,14]]
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [[1,3,5,6,9,11,13],[2,4,7,8,10,12,14]] => [[1,2,4,7,8,10,12],[3,5,6,9,11,13,14]]
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [[1,3,5,6,9,11,12],[2,4,7,8,10,13,14]] => [[1,2,4,7,8,10,13],[3,5,6,9,11,12,14]]
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [[1,3,5,6,9,10,13],[2,4,7,8,11,12,14]] => [[1,2,4,7,8,11,12],[3,5,6,9,10,13,14]]
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [[1,3,5,6,9,10,11],[2,4,7,8,12,13,14]] => [[1,2,4,7,8,11,13],[3,5,6,9,10,12,14]]
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [[1,3,5,6,7,11,13],[2,4,8,9,10,12,14]] => [[1,2,4,7,9,10,12],[3,5,6,8,11,13,14]]
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [[1,3,5,6,7,11,12],[2,4,8,9,10,13,14]] => [[1,2,4,7,9,10,13],[3,5,6,8,11,12,14]]
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [[1,3,5,6,7,8,13],[2,4,9,10,11,12,14]] => [[1,2,4,7,9,11,12],[3,5,6,8,10,13,14]]
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [[1,3,5,6,7,8,9],[2,4,10,11,12,13,14]] => [[1,2,4,7,9,11,13],[3,5,6,8,10,12,14]]
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [[1,3,4,7,9,11,13],[2,5,6,8,10,12,14]] => [[1,2,5,6,8,10,12],[3,4,7,9,11,13,14]]
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [[1,3,4,7,9,11,12],[2,5,6,8,10,13,14]] => [[1,2,5,6,8,10,13],[3,4,7,9,11,12,14]]
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [[1,3,4,7,9,10,13],[2,5,6,8,11,12,14]] => [[1,2,5,6,8,11,12],[3,4,7,9,10,13,14]]
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [[1,3,4,7,9,10,11],[2,5,6,8,12,13,14]] => [[1,2,5,6,8,11,13],[3,4,7,9,10,12,14]]
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [[1,3,4,7,8,11,13],[2,5,6,9,10,12,14]] => [[1,2,5,6,9,10,12],[3,4,7,8,11,13,14]]
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [[1,3,4,7,8,11,12],[2,5,6,9,10,13,14]] => [[1,2,5,6,9,10,13],[3,4,7,8,11,12,14]]
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [[1,3,4,7,8,9,13],[2,5,6,10,11,12,14]] => [[1,2,5,6,9,11,12],[3,4,7,8,10,13,14]]
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [[1,3,4,7,8,9,10],[2,5,6,11,12,13,14]] => [[1,2,5,6,9,11,13],[3,4,7,8,10,12,14]]
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => [[1,3,4,5,9,11,13],[2,6,7,8,10,12,14]] => [[1,2,5,7,8,10,12],[3,4,6,9,11,13,14]]
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [[1,3,4,5,9,11,12],[2,6,7,8,10,13,14]] => [[1,2,5,7,8,10,13],[3,4,6,9,11,12,14]]
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [[1,3,4,5,9,10,13],[2,6,7,8,11,12,14]] => [[1,2,5,7,8,11,12],[3,4,6,9,10,13,14]]
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [[1,3,4,5,9,10,11],[2,6,7,8,12,13,14]] => [[1,2,5,7,8,11,13],[3,4,6,9,10,12,14]]
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => [[1,3,4,5,6,11,13],[2,7,8,9,10,12,14]] => [[1,2,5,7,9,10,12],[3,4,6,8,11,13,14]]
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => [[1,3,4,5,6,11,12],[2,7,8,9,10,13,14]] => [[1,2,5,7,9,10,13],[3,4,6,8,11,12,14]]
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => [[1,3,4,5,6,7,13],[2,8,9,10,11,12,14]] => [[1,2,5,7,9,11,12],[3,4,6,8,10,13,14]]
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [[1,3,4,5,6,7,8],[2,9,10,11,12,13,14]] => [[1,2,5,7,9,11,13],[3,4,6,8,10,12,14]]
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => [[1,2,5,7,9,11,13],[3,4,6,8,10,12,14]] => [[1,3,4,6,8,10,12],[2,5,7,9,11,13,14]]
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => [[1,2,5,7,9,11,12],[3,4,6,8,10,13,14]] => [[1,3,4,6,8,10,13],[2,5,7,9,11,12,14]]
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => [[1,2,5,7,9,10,13],[3,4,6,8,11,12,14]] => [[1,3,4,6,8,11,12],[2,5,7,9,10,13,14]]
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => [[1,2,5,7,9,10,11],[3,4,6,8,12,13,14]] => [[1,3,4,6,8,11,13],[2,5,7,9,10,12,14]]
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => [[1,2,5,7,8,11,13],[3,4,6,9,10,12,14]] => [[1,3,4,6,9,10,12],[2,5,7,8,11,13,14]]
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => [[1,2,5,7,8,11,12],[3,4,6,9,10,13,14]] => [[1,3,4,6,9,10,13],[2,5,7,8,11,12,14]]
>>> Load all 222 entries. <<<Map
bounce path
Description
The bounce path determined by an integer composition.
Map
to two-row standard tableau
Description
Return a standard tableau of shape $(n,n)$ where $n$ is the semilength of the Dyck path.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
Given a Dyck path $D$, its image is given by recording the positions of the up-steps in the first row and the positions of the down-steps in the second row.
Map
row-to-column-descents
Description
Return a standard tableau whose column descent set equals the row descent set of the original tableau.
A column descent in a standard tableau is an entry $i$ such that $i+1$ appears in a column to the left of the cell containing $i$, in English notation.
A row descent is an entry $i$ such that $i+1$ appears in a row above of the cell containing $i$.
A column descent in a standard tableau is an entry $i$ such that $i+1$ appears in a column to the left of the cell containing $i$, in English notation.
A row descent is an entry $i$ such that $i+1$ appears in a row above of the cell containing $i$.
searching the database
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