Euler's partition theorem and the combinatorics of -sequences

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Define a q-analog of the l-nomial: a n (l) (q) = un l − v l nqn u l − v l q ; ∆(l) n (q) = ul n + v l n qn [ n k ] (l) q = a(l) n (q) a (l) n−1 (q) · · · a(l) n−k+1 (q) a (l) (q) a(l) (q) · · · a(l) 1 (q) . k k−1 Then [ n k ] (l) q = q k [ n − 2 k ] (l) q [ ] + ∆ (l) n − 2 (l) n−1 (q) k − 1 q [ ] n − 2 ( + q n−k k − 2 q and n∑ [ n k k=0 ] (l) q q k(k+1)/2 z k = α n (q, z) ⌊n/2⌋ ∏ i=1 (1 + zq i ∆ (l) n−2i+1 (q) + z2 q n+1 ). where α n (q, z) = 1 if n is even; (1 + zq (n+1)/2 ) if n is odd.
Proofs can be derived from this partition-theoretic interpretation:
Page 1 and 2:
Euler’s partition theorem and the
Page 3 and 4:
Overview Euler’s partition theore
Page 5 and 6:
Overview 1, 2, 3, . . . l-sequences
Page 7 and 8: Overview 1, 2, 3, . . . l-sequences
Page 9 and 10: Sylvester’s Bijection
Page 11 and 12: l-sequences For integer l ≥ 2, de
Page 13 and 14: l-sequences For integer l ≥ 2, de
Page 15 and 16: l = 2 The l-Euler theorem [BME2]: T
Page 17 and 18: l = 3 The l-Euler theorem [BME2]: T
Page 19: The insertion step To insert a k +
Page 22 and 23: Binary numeration system 1 0 1 1 0
Page 24 and 25: Binary numeration system 1 0 1 1 0
Page 26: Theorem [Fraenkel 1985] Every nonne
Page 31 and 32: Lecture Hall Partitions
Page 33 and 34: The Lecture Hall Theorem [BME1] The
Page 35 and 36: Θ (l) n : Bijection for the l-Lect
Page 44 and 45: Truncated lecture hall partitions L
Page 46 and 47: Theorem [Corteel,S 2004] Given posi
Page 48 and 49: The l-nomial coefficient Example (
Page 50 and 51: Let u l and v l be the roots of the
Page 52 and 53: Let u l and v l be the roots of the
Page 54 and 55: An l-nomial theorem [LS]: An analog
Page 56 and 57: A coin-flipping interpretation of t
Page 60 and 61: Another q-analog of the l-nomial Le
Page 62 and 63: Question: When l = 2, several refin
Page 64 and 65: Question: What is the generating fu
Page 66 and 67: CanaDAM 2009 2nd Canadian Discrete
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Euler's partition theorem and the combinatorics of -sequences

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