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The Plancherel measure and the partial fraction expansion. 2. Recall that we denote by X I , x2,. . , xd the points of minima, and by yl, . . , yd-1, the points of maxima of a diagram v = w(u). Condition (3) is immaterial for what follows, so we drop it. A function v = w(u) that satisfies only the two conditions (1) wl(u) = &I, and (2) there exists c E R such that w(u) = lu - cl for sufficiently large / u ( , will be called a rectangular diagram. The point c = C x k - C yk is called the centre of the diagram, and the number the area of the diagram.

Corresponding to w, = (1 - q7') / (1 - t n ) ) : (SO) If X = ( X I , . . ,An) is a Young diagram with n nonzero rows and :r = (21,. . , x,, 0 , . 5) PA(x; w) = 21 2 2 . . 2, PA* (2;w), where A, = (A1 - 1 , . . , A n - 1). We show that this property does not hold for more general sequences w. T H E O R E20 M ([119], Theorem 2). 9) wn=(Y, wkm = a , TLXl,2, . . , W, = 1 if n $ 0 mod rn. 7). The latter are obtained as follows. Substitute two variables x = ( x l , 2 2 ) into Px(x; q, t ) , then set 21x2 = 1, and consider PAas a polynomial in one variable y = xl r 2 .

An) is a Young diagram with n nonzero rows and :r = (21,. . , x,, 0 , . 5) PA(x; w) = 21 2 2 . . 2, PA* (2;w), where A, = (A1 - 1 , . . , A n - 1). We show that this property does not hold for more general sequences w. T H E O R E20 M ([119], Theorem 2). 9) wn=(Y, wkm = a , TLXl,2, . . , W, = 1 if n $ 0 mod rn. 7). The latter are obtained as follows. Substitute two variables x = ( x l , 2 2 ) into Px(x; q, t ) , then set 21x2 = 1, and consider PAas a polynomial in one variable y = xl r 2 .