Euler's partition theorem and the combinatorics of -sequences

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Let {a n } = {a (l) n }. The l-Lecture Hall Theorem [BME2]: The generating function for integer sequences λ 1 , λ 2 , . . . , λ n satisfying: L (l) n : λ 1 a n ≥ λ 2 a n−1 ≥ . . . ≥ λ n−1 a 2 ≥ λ n a 1 ≥ 0 is L (l) n (q) = n∏ i=1 1 (1 − q a i−1 +a i ) lim n→∞ (l-Lecture Hall Theorem) = l-Euler Theorem since as n → ∞, a n /a n−1 → c l
Θ (l) n : Bijection for the l-Lecture Hall Theorem Given a partition µ into parts in {a 0 + a 1 , a 1 + a 2 , a 2 + a 3 , . . . , a n−1 + a n } construct λ = (λ 1 , λ 2 , . . . , λ n ) by inserting the parts of µ in nonincreasing order as follows: To insert a k−1 + a k into (λ 1 , λ 2 , . . . , λ n ): If k = 1, then add a 1 to λ 1 ; otherwise, if (λ 1 + a k − a k−1 ) ≥ (a n /a n−1 )(λ 2 + a k−1 − a k−2 ), add a k − a k−1 to λ 1 , add a k−1 − a k−2 to λ 2 ; recursively insert a k−2 + a k−1 into (λ 3 , λ 4 , . . . , λ n ) otherwise, add a k to λ 1 , and add a k−1 to λ 2 .
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: The Lecture Hall Theorem [BME1] The
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 58 and 59: Define a q-analog of the l-nomial:
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

Euler's partition theorem and the combinatorics of -sequences

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