We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions.
M. Barnabei, F. Bonetti, M. Silimbani (2011). Restricted involutions and Motzkin paths. ADVANCES IN APPLIED MATHEMATICS, 47, 102-115 [10.1016/j.aam.2010.05.002].
Restricted involutions and Motzkin paths
BARNABEI, MARILENA;BONETTI, FLAVIO;SILIMBANI, MATTEO
2011
Abstract
We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
