Discrete Mathematics 49 (1984) 41-44 41 North-Holland SOME GENERALIZED DURFEE SQUARE IDENTITIF~ Ira M. GESSEL* Dept. of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Received 6 March 1981 Revised 9 February 1983 A generalized q-binomial Vandermonde convolution of Sulanke is proved using a generaliza- tion of the Durfee square of a partition. Bender [4] proved the following generalization of the q-binomial Vandermonde convolution: k=o LkJl_ n-k J n ' where for l~<i<n, A~+I-Ai is 0 or 1, and the q-binomial coefficient [~ is defined for integers n and all real x by [~] = 0 for n <0, [~] = 1, and for n > O, "-~ (q~-' - 1) (2) Bender's identity was generalized by Evans [5], who Obtained a formula for the difference between the sides of (1) when the restriction on A~÷I-A~ is removed. Sulanke [7] generalized Bender's identity in a different direction, using lattice paths. We prove here a more symmetrical version of Sulanke's identity, using a generalization of the concept of the Duffee square of a partition. Although our proof is ultimately equivalent to Sulanke's, the connection with partitions suggests further generalization and applications. A partition ~ of a nonnegative integer l is a (possibly empty) sequence )t1~>)t2>~"" "~>Xk Of positive integers with sum l=lxl. The Ferrets graph of )t is the set of all pairs (i, ]) of integers with 1 ~< i ~< k and 1 ~<] ~<~. The points of the Ferrers graph are displayed like entries of a matrix; thus the Ferrers graph of 421 is The Durfee square of a partition ,k is the largest square which fits in the upper * Research partially supported by NSF Grant MCS 8105188 0012-365X/84/$3.00 © 1984, Elsevier Science Publishers B.V. (North-Holland)