Identifier
Mp00142: Dyck paths promotionDyck paths
Mp00035: Dyck paths to alternating sign matrixAlternating sign matrices
Mp00098: Alternating sign matrices link pattern Perfect matchings
Images
=>
Cc0005;cc-rep-0Cc0005;cc-rep-1Cc0017;cc-rep-2Cc0012;cc-rep-3
[1,0]=>[1,0]=>[[1]]=>[(1,2)] [1,0,1,0]=>[1,1,0,0]=>[[0,1],[1,0]]=>[(1,2),(3,4)] [1,1,0,0]=>[1,0,1,0]=>[[1,0],[0,1]]=>[(1,4),(2,3)] [1,0,1,0,1,0]=>[1,1,0,1,0,0]=>[[0,1,0],[1,-1,1],[0,1,0]]=>[(1,2),(3,6),(4,5)] [1,0,1,1,0,0]=>[1,1,0,0,1,0]=>[[0,1,0],[1,0,0],[0,0,1]]=>[(1,2),(3,4),(5,6)] [1,1,0,0,1,0]=>[1,1,1,0,0,0]=>[[0,0,1],[1,0,0],[0,1,0]]=>[(1,6),(2,3),(4,5)] [1,1,0,1,0,0]=>[1,0,1,0,1,0]=>[[1,0,0],[0,1,0],[0,0,1]]=>[(1,6),(2,5),(3,4)] [1,1,1,0,0,0]=>[1,0,1,1,0,0]=>[[1,0,0],[0,0,1],[0,1,0]]=>[(1,6),(2,3),(4,5)] [1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,0]=>[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]=>[(1,2),(3,8),(4,7),(5,6)] [1,0,1,0,1,1,0,0]=>[1,1,0,1,0,0,1,0]=>[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]=>[(1,2),(3,8),(4,5),(6,7)] [1,0,1,1,0,0,1,0]=>[1,1,0,1,1,0,0,0]=>[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]=>[(1,2),(3,4),(5,6),(7,8)] [1,0,1,1,0,1,0,0]=>[1,1,0,0,1,0,1,0]=>[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]=>[(1,2),(3,6),(4,5),(7,8)] [1,0,1,1,1,0,0,0]=>[1,1,0,0,1,1,0,0]=>[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]=>[(1,2),(3,4),(5,6),(7,8)] [1,1,0,0,1,0,1,0]=>[1,1,1,0,0,1,0,0]=>[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]=>[(1,8),(2,3),(4,7),(5,6)] [1,1,0,0,1,1,0,0]=>[1,1,1,0,0,0,1,0]=>[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]=>[(1,8),(2,3),(4,5),(6,7)] [1,1,0,1,0,0,1,0]=>[1,1,1,0,1,0,0,0]=>[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]=>[(1,8),(2,7),(3,4),(5,6)] [1,1,0,1,0,1,0,0]=>[1,0,1,0,1,0,1,0]=>[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]=>[(1,8),(2,7),(3,6),(4,5)] [1,1,0,1,1,0,0,0]=>[1,0,1,0,1,1,0,0]=>[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]=>[(1,8),(2,7),(3,4),(5,6)] [1,1,1,0,0,0,1,0]=>[1,1,1,1,0,0,0,0]=>[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]=>[(1,8),(2,3),(4,7),(5,6)] [1,1,1,0,0,1,0,0]=>[1,0,1,1,0,0,1,0]=>[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]=>[(1,8),(2,3),(4,5),(6,7)] [1,1,1,0,1,0,0,0]=>[1,0,1,1,0,1,0,0]=>[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]=>[(1,8),(2,3),(4,7),(5,6)] [1,1,1,1,0,0,0,0]=>[1,0,1,1,1,0,0,0]=>[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]=>[(1,8),(2,7),(3,4),(5,6)] [1,0,1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,1,0,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[(1,2),(3,10),(4,9),(5,8),(6,7)] [1,0,1,0,1,0,1,1,0,0]=>[1,1,0,1,0,1,0,0,1,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[(1,2),(3,10),(4,9),(5,6),(7,8)] [1,0,1,0,1,1,0,0,1,0]=>[1,1,0,1,0,1,1,0,0,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,2),(3,10),(4,5),(6,7),(8,9)] [1,0,1,0,1,1,0,1,0,0]=>[1,1,0,1,0,0,1,0,1,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[(1,2),(3,10),(4,7),(5,6),(8,9)] [1,0,1,0,1,1,1,0,0,0]=>[1,1,0,1,0,0,1,1,0,0]=>[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[(1,2),(3,10),(4,5),(6,7),(8,9)] [1,0,1,1,0,0,1,0,1,0]=>[1,1,0,1,1,0,0,1,0,0]=>[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[(1,2),(3,4),(5,8),(6,7),(9,10)] [1,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,1,0,0,0,1,0]=>[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[(1,2),(3,4),(5,6),(7,8),(9,10)] [1,0,1,1,0,1,0,0,1,0]=>[1,1,0,1,1,0,1,0,0,0]=>[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,2),(3,8),(4,5),(6,7),(9,10)] [1,0,1,1,0,1,0,1,0,0]=>[1,1,0,0,1,0,1,0,1,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[(1,2),(3,8),(4,7),(5,6),(9,10)] [1,0,1,1,0,1,1,0,0,0]=>[1,1,0,0,1,0,1,1,0,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[(1,2),(3,8),(4,5),(6,7),(9,10)] [1,0,1,1,1,0,0,0,1,0]=>[1,1,0,1,1,1,0,0,0,0]=>[[0,1,0,0,0],[1,-1,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,2),(3,4),(5,8),(6,7),(9,10)] [1,0,1,1,1,0,0,1,0,0]=>[1,1,0,0,1,1,0,0,1,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[(1,2),(3,4),(5,6),(7,8),(9,10)] [1,0,1,1,1,0,1,0,0,0]=>[1,1,0,0,1,1,0,1,0,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[(1,2),(3,4),(5,8),(6,7),(9,10)] [1,0,1,1,1,1,0,0,0,0]=>[1,1,0,0,1,1,1,0,0,0]=>[[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,2),(3,8),(4,5),(6,7),(9,10)] [1,1,0,0,1,0,1,0,1,0]=>[1,1,1,0,0,1,0,1,0,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[(1,10),(2,3),(4,9),(5,8),(6,7)] [1,1,0,0,1,0,1,1,0,0]=>[1,1,1,0,0,1,0,0,1,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[(1,10),(2,3),(4,9),(5,6),(7,8)] [1,1,0,0,1,1,0,0,1,0]=>[1,1,1,0,0,1,1,0,0,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,3),(4,5),(6,7),(8,9)] [1,1,0,0,1,1,0,1,0,0]=>[1,1,1,0,0,0,1,0,1,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[(1,10),(2,3),(4,7),(5,6),(8,9)] [1,1,0,0,1,1,1,0,0,0]=>[1,1,1,0,0,0,1,1,0,0]=>[[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[(1,10),(2,3),(4,5),(6,7),(8,9)] [1,1,0,1,0,0,1,0,1,0]=>[1,1,1,0,1,0,0,1,0,0]=>[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[(1,10),(2,9),(3,4),(5,8),(6,7)] [1,1,0,1,0,0,1,1,0,0]=>[1,1,1,0,1,0,0,0,1,0]=>[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[(1,10),(2,9),(3,4),(5,6),(7,8)] [1,1,0,1,0,1,0,0,1,0]=>[1,1,1,0,1,0,1,0,0,0]=>[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,9),(3,8),(4,5),(6,7)] [1,1,0,1,0,1,0,1,0,0]=>[1,0,1,0,1,0,1,0,1,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[(1,10),(2,9),(3,8),(4,7),(5,6)] [1,1,0,1,0,1,1,0,0,0]=>[1,0,1,0,1,0,1,1,0,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[(1,10),(2,9),(3,8),(4,5),(6,7)] [1,1,0,1,1,0,0,0,1,0]=>[1,1,1,0,1,1,0,0,0,0]=>[[0,0,1,0,0],[1,0,-1,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,9),(3,4),(5,8),(6,7)] [1,1,0,1,1,0,0,1,0,0]=>[1,0,1,0,1,1,0,0,1,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[(1,10),(2,9),(3,4),(5,6),(7,8)] [1,1,0,1,1,0,1,0,0,0]=>[1,0,1,0,1,1,0,1,0,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[(1,10),(2,9),(3,4),(5,8),(6,7)] [1,1,0,1,1,1,0,0,0,0]=>[1,0,1,0,1,1,1,0,0,0]=>[[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,9),(3,8),(4,5),(6,7)] [1,1,1,0,0,0,1,0,1,0]=>[1,1,1,1,0,0,0,1,0,0]=>[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[(1,10),(2,3),(4,9),(5,8),(6,7)] [1,1,1,0,0,0,1,1,0,0]=>[1,1,1,1,0,0,0,0,1,0]=>[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[(1,10),(2,3),(4,9),(5,6),(7,8)] [1,1,1,0,0,1,0,0,1,0]=>[1,1,1,1,0,0,1,0,0,0]=>[[0,0,0,1,0],[1,0,0,0,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,3),(4,5),(6,7),(8,9)] [1,1,1,0,0,1,0,1,0,0]=>[1,0,1,1,0,0,1,0,1,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]=>[(1,10),(2,3),(4,7),(5,6),(8,9)] [1,1,1,0,0,1,1,0,0,0]=>[1,0,1,1,0,0,1,1,0,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]=>[(1,10),(2,3),(4,5),(6,7),(8,9)] [1,1,1,0,1,0,0,0,1,0]=>[1,1,1,1,0,1,0,0,0,0]=>[[0,0,0,1,0],[1,0,0,-1,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,3),(4,9),(5,8),(6,7)] [1,1,1,0,1,0,0,1,0,0]=>[1,0,1,1,0,1,0,0,1,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]=>[(1,10),(2,3),(4,9),(5,6),(7,8)] [1,1,1,0,1,0,1,0,0,0]=>[1,0,1,1,0,1,0,1,0,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[(1,10),(2,3),(4,9),(5,8),(6,7)] [1,1,1,0,1,1,0,0,0,0]=>[1,0,1,1,0,1,1,0,0,0]=>[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,3),(4,5),(6,7),(8,9)] [1,1,1,1,0,0,0,0,1,0]=>[1,1,1,1,1,0,0,0,0,0]=>[[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,9),(3,4),(5,8),(6,7)] [1,1,1,1,0,0,0,1,0,0]=>[1,0,1,1,1,0,0,0,1,0]=>[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]=>[(1,10),(2,9),(3,4),(5,6),(7,8)] [1,1,1,1,0,0,1,0,0,0]=>[1,0,1,1,1,0,0,1,0,0]=>[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]=>[(1,10),(2,9),(3,4),(5,8),(6,7)] [1,1,1,1,0,1,0,0,0,0]=>[1,0,1,1,1,0,1,0,0,0]=>[[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,9),(3,8),(4,5),(6,7)] [1,1,1,1,1,0,0,0,0,0]=>[1,0,1,1,1,1,0,0,0,0]=>[[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]=>[(1,10),(2,9),(3,4),(5,8),(6,7)] [1,0,1,0,1,0,1,0,1,0,1,0]=>[1,1,0,1,0,1,0,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)] [1,0,1,0,1,0,1,0,1,1,0,0]=>[1,1,0,1,0,1,0,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,2),(3,12),(4,11),(5,10),(6,7),(8,9)] [1,0,1,0,1,0,1,1,0,0,1,0]=>[1,1,0,1,0,1,0,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)] [1,0,1,0,1,0,1,1,0,1,0,0]=>[1,1,0,1,0,1,0,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,2),(3,12),(4,11),(5,8),(6,7),(9,10)] [1,0,1,0,1,0,1,1,1,0,0,0]=>[1,1,0,1,0,1,0,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,2),(3,12),(4,11),(5,6),(7,8),(9,10)] [1,0,1,0,1,1,0,0,1,0,1,0]=>[1,1,0,1,0,1,1,0,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)] [1,0,1,0,1,1,0,0,1,1,0,0]=>[1,1,0,1,0,1,1,0,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)] [1,0,1,0,1,1,0,1,0,0,1,0]=>[1,1,0,1,0,1,1,0,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)] [1,0,1,0,1,1,0,1,0,1,0,0]=>[1,1,0,1,0,0,1,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,2),(3,12),(4,9),(5,8),(6,7),(10,11)] [1,0,1,0,1,1,0,1,1,0,0,0]=>[1,1,0,1,0,0,1,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)] [1,0,1,0,1,1,1,0,0,0,1,0]=>[1,1,0,1,0,1,1,1,0,0,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)] [1,0,1,0,1,1,1,0,0,1,0,0]=>[1,1,0,1,0,0,1,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,2),(3,12),(4,5),(6,7),(8,9),(10,11)] [1,0,1,0,1,1,1,0,1,0,0,0]=>[1,1,0,1,0,0,1,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,2),(3,12),(4,5),(6,9),(7,8),(10,11)] [1,0,1,0,1,1,1,1,0,0,0,0]=>[1,1,0,1,0,0,1,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,12),(4,9),(5,6),(7,8),(10,11)] [1,0,1,1,0,0,1,0,1,0,1,0]=>[1,1,0,1,1,0,0,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] [1,0,1,1,0,0,1,0,1,1,0,0]=>[1,1,0,1,1,0,0,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)] [1,0,1,1,0,0,1,1,0,0,1,0]=>[1,1,0,1,1,0,0,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] [1,0,1,1,0,0,1,1,0,1,0,0]=>[1,1,0,1,1,0,0,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)] [1,0,1,1,0,0,1,1,1,0,0,0]=>[1,1,0,1,1,0,0,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] [1,0,1,1,0,1,0,0,1,0,1,0]=>[1,1,0,1,1,0,1,0,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)] [1,0,1,1,0,1,0,0,1,1,0,0]=>[1,1,0,1,1,0,1,0,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)] [1,0,1,1,0,1,0,1,0,0,1,0]=>[1,1,0,1,1,0,1,0,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)] [1,0,1,1,0,1,0,1,0,1,0,0]=>[1,1,0,0,1,0,1,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)] [1,0,1,1,0,1,0,1,1,0,0,0]=>[1,1,0,0,1,0,1,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)] [1,0,1,1,0,1,1,0,0,0,1,0]=>[1,1,0,1,1,0,1,1,0,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)] [1,0,1,1,0,1,1,0,0,1,0,0]=>[1,1,0,0,1,0,1,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)] [1,0,1,1,0,1,1,0,1,0,0,0]=>[1,1,0,0,1,0,1,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)] [1,0,1,1,0,1,1,1,0,0,0,0]=>[1,1,0,0,1,0,1,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)] [1,0,1,1,1,0,0,0,1,0,1,0]=>[1,1,0,1,1,1,0,0,0,1,0,0]=>[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] [1,0,1,1,1,0,0,0,1,1,0,0]=>[1,1,0,1,1,1,0,0,0,0,1,0]=>[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)] [1,0,1,1,1,0,0,1,0,0,1,0]=>[1,1,0,1,1,1,0,0,1,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] [1,0,1,1,1,0,0,1,0,1,0,0]=>[1,1,0,0,1,1,0,0,1,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)] [1,0,1,1,1,0,0,1,1,0,0,0]=>[1,1,0,0,1,1,0,0,1,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] [1,0,1,1,1,0,1,0,0,0,1,0]=>[1,1,0,1,1,1,0,1,0,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] [1,0,1,1,1,0,1,0,0,1,0,0]=>[1,1,0,0,1,1,0,1,0,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)] [1,0,1,1,1,0,1,0,1,0,0,0]=>[1,1,0,0,1,1,0,1,0,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)] [1,0,1,1,1,0,1,1,0,0,0,0]=>[1,1,0,0,1,1,0,1,1,0,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)] [1,0,1,1,1,1,0,0,0,0,1,0]=>[1,1,0,1,1,1,1,0,0,0,0,0]=>[[0,1,0,0,0,0],[1,-1,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)] [1,0,1,1,1,1,0,0,0,1,0,0]=>[1,1,0,0,1,1,1,0,0,0,1,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,2),(3,10),(4,5),(6,7),(8,9),(11,12)] [1,0,1,1,1,1,0,0,1,0,0,0]=>[1,1,0,0,1,1,1,0,0,1,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)] [1,0,1,1,1,1,0,1,0,0,0,0]=>[1,1,0,0,1,1,1,0,1,0,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,9),(5,6),(7,8),(11,12)] [1,0,1,1,1,1,1,0,0,0,0,0]=>[1,1,0,0,1,1,1,1,0,0,0,0]=>[[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,2),(3,10),(4,5),(6,9),(7,8),(11,12)] [1,1,0,0,1,0,1,0,1,0,1,0]=>[1,1,1,0,0,1,0,1,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)] [1,1,0,0,1,0,1,0,1,1,0,0]=>[1,1,1,0,0,1,0,1,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)] [1,1,0,0,1,0,1,1,0,0,1,0]=>[1,1,1,0,0,1,0,1,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)] [1,1,0,0,1,0,1,1,0,1,0,0]=>[1,1,1,0,0,1,0,0,1,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)] [1,1,0,0,1,0,1,1,1,0,0,0]=>[1,1,1,0,0,1,0,0,1,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)] [1,1,0,0,1,1,0,0,1,0,1,0]=>[1,1,1,0,0,1,1,0,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)] [1,1,0,0,1,1,0,0,1,1,0,0]=>[1,1,1,0,0,1,1,0,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)] [1,1,0,0,1,1,0,1,0,0,1,0]=>[1,1,1,0,0,1,1,0,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)] [1,1,0,0,1,1,0,1,0,1,0,0]=>[1,1,1,0,0,0,1,0,1,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)] [1,1,0,0,1,1,0,1,1,0,0,0]=>[1,1,1,0,0,0,1,0,1,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)] [1,1,0,0,1,1,1,0,0,0,1,0]=>[1,1,1,0,0,1,1,1,0,0,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)] [1,1,0,0,1,1,1,0,0,1,0,0]=>[1,1,1,0,0,0,1,1,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)] [1,1,0,0,1,1,1,0,1,0,0,0]=>[1,1,1,0,0,0,1,1,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)] [1,1,0,0,1,1,1,1,0,0,0,0]=>[1,1,1,0,0,0,1,1,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)] [1,1,0,1,0,0,1,0,1,0,1,0]=>[1,1,1,0,1,0,0,1,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,0,1,0,0,1,0,1,1,0,0]=>[1,1,1,0,1,0,0,1,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)] [1,1,0,1,0,0,1,1,0,0,1,0]=>[1,1,1,0,1,0,0,1,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)] [1,1,0,1,0,0,1,1,0,1,0,0]=>[1,1,1,0,1,0,0,0,1,0,1,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,4),(5,8),(6,7),(9,10)] [1,1,0,1,0,0,1,1,1,0,0,0]=>[1,1,1,0,1,0,0,0,1,1,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)] [1,1,0,1,0,1,0,0,1,0,1,0]=>[1,1,1,0,1,0,1,0,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,1,0,1,0,1,0,0,1,1,0,0]=>[1,1,1,0,1,0,1,0,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)] [1,1,0,1,0,1,0,1,0,0,1,0]=>[1,1,1,0,1,0,1,0,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)] [1,1,0,1,0,1,0,1,0,1,0,0]=>[1,0,1,0,1,0,1,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)] [1,1,0,1,0,1,0,1,1,0,0,0]=>[1,0,1,0,1,0,1,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)] [1,1,0,1,0,1,1,0,0,0,1,0]=>[1,1,1,0,1,0,1,1,0,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,1,0,1,0,1,1,0,0,1,0,0]=>[1,0,1,0,1,0,1,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)] [1,1,0,1,0,1,1,0,1,0,0,0]=>[1,0,1,0,1,0,1,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,1,0,1,0,1,1,1,0,0,0,0]=>[1,0,1,0,1,0,1,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)] [1,1,0,1,1,0,0,0,1,0,1,0]=>[1,1,1,0,1,1,0,0,0,1,0,0]=>[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,0,1,1,0,0,0,1,1,0,0]=>[1,1,1,0,1,1,0,0,0,0,1,0]=>[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)] [1,1,0,1,1,0,0,1,0,0,1,0]=>[1,1,1,0,1,1,0,0,1,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)] [1,1,0,1,1,0,0,1,0,1,0,0]=>[1,0,1,0,1,1,0,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,4),(5,8),(6,7),(9,10)] [1,1,0,1,1,0,0,1,1,0,0,0]=>[1,0,1,0,1,1,0,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)] [1,1,0,1,1,0,1,0,0,0,1,0]=>[1,1,1,0,1,1,0,1,0,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,0,1,1,0,1,0,0,1,0,0]=>[1,0,1,0,1,1,0,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)] [1,1,0,1,1,0,1,0,1,0,0,0]=>[1,0,1,0,1,1,0,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,0,1,1,0,1,1,0,0,0,0]=>[1,0,1,0,1,1,0,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)] [1,1,0,1,1,1,0,0,0,0,1,0]=>[1,1,1,0,1,1,1,0,0,0,0,0]=>[[0,0,1,0,0,0],[1,0,-1,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,1,0,1,1,1,0,0,0,1,0,0]=>[1,0,1,0,1,1,1,0,0,0,1,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)] [1,1,0,1,1,1,0,0,1,0,0,0]=>[1,0,1,0,1,1,1,0,0,1,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,1,0,1,1,1,0,1,0,0,0,0]=>[1,0,1,0,1,1,1,0,1,0,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)] [1,1,0,1,1,1,1,0,0,0,0,0]=>[1,0,1,0,1,1,1,1,0,0,0,0]=>[[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,1,1,0,0,0,1,0,1,0,1,0]=>[1,1,1,1,0,0,0,1,0,1,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)] [1,1,1,0,0,0,1,0,1,1,0,0]=>[1,1,1,1,0,0,0,1,0,0,1,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)] [1,1,1,0,0,0,1,1,0,0,1,0]=>[1,1,1,1,0,0,0,1,1,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)] [1,1,1,0,0,0,1,1,0,1,0,0]=>[1,1,1,1,0,0,0,0,1,0,1,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)] [1,1,1,0,0,0,1,1,1,0,0,0]=>[1,1,1,1,0,0,0,0,1,1,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)] [1,1,1,0,0,1,0,0,1,0,1,0]=>[1,1,1,1,0,0,1,0,0,1,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)] [1,1,1,0,0,1,0,0,1,1,0,0]=>[1,1,1,1,0,0,1,0,0,0,1,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)] [1,1,1,0,0,1,0,1,0,0,1,0]=>[1,1,1,1,0,0,1,0,1,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)] [1,1,1,0,0,1,0,1,0,1,0,0]=>[1,0,1,1,0,0,1,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,9),(5,8),(6,7),(10,11)] [1,1,1,0,0,1,0,1,1,0,0,0]=>[1,0,1,1,0,0,1,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)] [1,1,1,0,0,1,1,0,0,0,1,0]=>[1,1,1,1,0,0,1,1,0,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)] [1,1,1,0,0,1,1,0,0,1,0,0]=>[1,0,1,1,0,0,1,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)] [1,1,1,0,0,1,1,0,1,0,0,0]=>[1,0,1,1,0,0,1,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)] [1,1,1,0,0,1,1,1,0,0,0,0]=>[1,0,1,1,0,0,1,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)] [1,1,1,0,1,0,0,0,1,0,1,0]=>[1,1,1,1,0,1,0,0,0,1,0,0]=>[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)] [1,1,1,0,1,0,0,0,1,1,0,0]=>[1,1,1,1,0,1,0,0,0,0,1,0]=>[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)] [1,1,1,0,1,0,0,1,0,0,1,0]=>[1,1,1,1,0,1,0,0,1,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)] [1,1,1,0,1,0,0,1,0,1,0,0]=>[1,0,1,1,0,1,0,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,11),(5,8),(6,7),(9,10)] [1,1,1,0,1,0,0,1,1,0,0,0]=>[1,0,1,1,0,1,0,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)] [1,1,1,0,1,0,1,0,0,0,1,0]=>[1,1,1,1,0,1,0,1,0,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,-1,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)] [1,1,1,0,1,0,1,0,0,1,0,0]=>[1,0,1,1,0,1,0,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,11),(5,10),(6,7),(8,9)] [1,1,1,0,1,0,1,0,1,0,0,0]=>[1,0,1,1,0,1,0,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,10),(6,9),(7,8)] [1,1,1,0,1,0,1,1,0,0,0,0]=>[1,0,1,1,0,1,0,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,11),(5,6),(7,8),(9,10)] [1,1,1,0,1,1,0,0,0,0,1,0]=>[1,1,1,1,0,1,1,0,0,0,0,0]=>[[0,0,0,1,0,0],[1,0,0,-1,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)] [1,1,1,0,1,1,0,0,0,1,0,0]=>[1,0,1,1,0,1,1,0,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,3),(4,5),(6,7),(8,9),(10,11)] [1,1,1,0,1,1,0,0,1,0,0,0]=>[1,0,1,1,0,1,1,0,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)] [1,1,1,0,1,1,0,1,0,0,0,0]=>[1,0,1,1,0,1,1,0,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,9),(5,6),(7,8),(10,11)] [1,1,1,0,1,1,1,0,0,0,0,0]=>[1,0,1,1,0,1,1,1,0,0,0,0]=>[[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,3),(4,5),(6,9),(7,8),(10,11)] [1,1,1,1,0,0,0,0,1,0,1,0]=>[1,1,1,1,1,0,0,0,0,1,0,0]=>[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,1,1,0,0,0,0,1,1,0,0]=>[1,1,1,1,1,0,0,0,0,0,1,0]=>[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)] [1,1,1,1,0,0,0,1,0,0,1,0]=>[1,1,1,1,1,0,0,0,1,0,0,0]=>[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)] [1,1,1,1,0,0,0,1,0,1,0,0]=>[1,0,1,1,1,0,0,0,1,0,1,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,4),(5,8),(6,7),(9,10)] [1,1,1,1,0,0,0,1,1,0,0,0]=>[1,0,1,1,1,0,0,0,1,1,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)] [1,1,1,1,0,0,1,0,0,0,1,0]=>[1,1,1,1,1,0,0,1,0,0,0,0]=>[[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,1,1,0,0,1,0,0,1,0,0]=>[1,0,1,1,1,0,0,1,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)] [1,1,1,1,0,0,1,0,1,0,0,0]=>[1,0,1,1,1,0,0,1,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,1,1,0,0,1,1,0,0,0,0]=>[1,0,1,1,1,0,0,1,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)] [1,1,1,1,0,1,0,0,0,0,1,0]=>[1,1,1,1,1,0,1,0,0,0,0,0]=>[[0,0,0,0,1,0],[1,0,0,0,-1,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,1,1,1,0,1,0,0,0,1,0,0]=>[1,0,1,1,1,0,1,0,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,10),(4,5),(6,7),(8,9)] [1,1,1,1,0,1,0,0,1,0,0,0]=>[1,0,1,1,1,0,1,0,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,1,1,1,0,1,0,1,0,0,0,0]=>[1,0,1,1,1,0,1,0,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,9),(5,6),(7,8)] [1,1,1,1,0,1,1,0,0,0,0,0]=>[1,0,1,1,1,0,1,1,0,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,-1,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,1,1,1,1,0,0,0,0,0,1,0]=>[1,1,1,1,1,1,0,0,0,0,0,0]=>[[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,1,1,1,0,0,0,0,1,0,0]=>[1,0,1,1,1,1,0,0,0,0,1,0]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]=>[(1,12),(2,11),(3,4),(5,10),(6,7),(8,9)] [1,1,1,1,1,0,0,0,1,0,0,0]=>[1,0,1,1,1,1,0,0,0,1,0,0]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,1,1,1,0,0,1,0,0,0,0]=>[1,0,1,1,1,1,0,0,1,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,-1,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,6),(7,8),(9,10)] [1,1,1,1,1,0,1,0,0,0,0,0]=>[1,0,1,1,1,1,0,1,0,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,-1,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,4),(5,10),(6,9),(7,8)] [1,1,1,1,1,1,0,0,0,0,0,0]=>[1,0,1,1,1,1,1,0,0,0,0,0]=>[[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]=>[(1,12),(2,11),(3,10),(4,5),(6,9),(7,8)] [1,0,1,1,1,0,0,1,1,0,0,1,1,0,0,0]=>[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]=>[[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]=>[(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16)] [1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]=>[1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]=>[[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,0,0,0,1],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0]]=>[(1,16),(2,15),(3,14),(4,13),(5,12),(6,7),(8,11),(9,10)]
Map
promotion
Description
The promotion of the two-row standard Young tableau of a Dyck path.
Dyck paths of semilength $n$ are in bijection with standard Young tableaux of shape $(n^2)$, see Mp00033to two-row standard tableau.
This map is the bijection on such standard Young tableaux given by Sch├╝tzenberger's promotion. For definitions and details, see [1] and the references therein.
Map
to alternating sign matrix
Description
Return the Dyck path as an alternating sign matrix.
This is an inclusion map from Dyck words of length $2n$ to certain
$n \times n$ alternating sign matrices.
Map
link pattern
Description
Sends an alternating sign matrix to the link pattern of the corresponding fully packed loop configuration.