By Harry Gonshor
The surreal numbers shape a process inclusive of either the normal actual numbers and the ordinals. in view that their advent by way of J. H. Conway, the idea of surreal numbers has visible a swift improvement revealing many average and fascinating homes. those notes supply a proper advent to the speculation in a transparent and lucid variety. The the writer is ready to lead the reader via to a couple of the issues within the box. the themes coated comprise exponentiation and generalized e-numbers.
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Extra resources for An Introduction to the Theory of Surreal Numbers
N] n−k 48 3. THE q, t-CATALAN NUMBERS n Proof. 10 holds for k = n. 46) = q (k−1)n n−k−1 [k] [n − k] k + u u(n−k) 2(n − k) − 2 − u q . 48) [k] 1 [k] 1 1 | n−k−1 = q (k−1)n | n−k−1 [n − k] (zq n−k )k+1 (z)n−k z [n − k] (z)n+1 z [k] n+n−k−1 . 1. 49) 2n 1 . [n + 1] n Proof. N. 51) n+1 2 q( )F n+1,1 (q, 1/q) 2(n + 1) − 2  2n  = [n + 1] n + 1 − 1 [n + 1] n n+1 n −n ( ) ) ( =q 2 Fn (q, 1/q) = q 2 Fn (q, 1/q). 53) corresponding to μ and μ . This also follows from the theorem that Cn (q, t) = H(DHn ; q, t).
Macdonald obtained the following two symmetry relations. 29) Kλ,μ (q, t) = Kλ ,μ (t, q). 31) ˜ λ,μ (1/q, 1/t). ˜ λ ,μ (q, t) = tn(μ) q n(μ ) K K Fischel [Fis95] ﬁrst obtained statistics for the case when μ has two columns. 31) this also implies statistics for the case where μ has two rows. Later Lapointe and Morse [LM03b] and Zabrocki [Zab98] independently found alternate descriptions of this case, but all of these are rather complicated to state. A simpler description of the two-column case based on the combinatorial formula for Macdonald polynomials is contained in Appendix A.
There is also a version of the shuﬄe conjecture for these spaces, which we describe at the end of Chapter 6. 18. In this exercise we prove the following alternate version of Koornwinder-Macdonald reciprocity, which is an unpublished result of the author. 73) a l x∈μ (1 − zq t ) = ˜ λ [1 − z − (1 − 1/q)(1 − 1/t)Bμ (1/q, 1/t); q, t] H . 74) tn(μ) 1 − 1/q a tl+1 1−z − (1 − q)Bλ (q, 1/t); 1/q, 1/t) . Pμ l /q a 1 − zt 1 − 1/t s∈μ (2) Use the symmetry relation Pμ (X; 1/q, 1/t) = Pμ (X; q, t) [Mac95, p.
An Introduction to the Theory of Surreal Numbers by Harry Gonshor