Geodesic
d-Regular Set Partitions and Rook Placements
Séminaire lotharingien de combinatoire, Tome 62 (2009-2010) 
Anisse Kasraoui. d-Regular Set Partitions and Rook Placements. Séminaire lotharingien de combinatoire, Tome 62 (2009-2010). http://geodesic.mathdoc.fr/item/SLC_2009-2010_62_a0/
@article{SLC_2009-2010_62_a0,
     author = {Anisse Kasraoui},
     title = {d-Regular {Set} {Partitions} and {Rook} {Placements}},
     journal = {S\'eminaire lotharingien de combinatoire},
     year = {2009-2010},
     volume = {62},
     url = {http://geodesic.mathdoc.fr/item/SLC_2009-2010_62_a0/}
}
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AU  - Anisse Kasraoui
TI  - d-Regular Set Partitions and Rook Placements
JO  - Séminaire lotharingien de combinatoire
PY  - 2009-2010
VL  - 62
UR  - http://geodesic.mathdoc.fr/item/SLC_2009-2010_62_a0/
ID  - SLC_2009-2010_62_a0
ER  - 
%0 Journal Article
%A Anisse Kasraoui
%T d-Regular Set Partitions and Rook Placements
%J Séminaire lotharingien de combinatoire
%D 2009-2010
%V 62
%U http://geodesic.mathdoc.fr/item/SLC_2009-2010_62_a0/
%F SLC_2009-2010_62_a0

Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

We use a classical correspondence between set partitions and rook placements on the triangular board to give a quick picture understanding of the "reduction identity"

|P(d)(n,k)| = |P(d-j)(n-j,k-j)|,

where P(d)(n,k) is the collection of all set partitions of [n]:={1,2,...,n} into k blocks such that for any two distinct elements x,y in the same block, we have |y-x| >= d. We also generalize an identity of Klazar on d-regular noncrossing partitions. Namely, we show that the number of d-regular l-noncrossing partitions of [n] is equal to the number of (d-1)-regular enhanced l-noncrossing partitions of [n-1].