Euler's partition theorem and the combinatorics of -sequences

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Binary numeration system 1 0 1 1 0 → 1 ∗ 2 4 + 0 ∗ 2 3 + 1 ∗ 2 2 + 1 ∗ 2 1 + 0 ∗ 2 0 = 22 Ternary numeration system (unique representation) 1 0 2 1 1 → 1 ∗ 3 4 + 0 ∗ 3 3 + 2 ∗ 3 2 + 1 ∗ 3 1 + 1 ∗ 3 0 = 138 A Fraenkel numeration system: ternary, but ... (unique representation) 1 0 2 1 1 → 1 ∗ 55 + 0 ∗ 21 + 2 ∗ 8 + 1 ∗ 3 + 1 ∗ 1 = 75 0 0 2 1 2 → 0 ∗ 55 + 0 ∗ 21 + 2 ∗ 8 + 1 ∗ 3 + 2 ∗ 1 = 21 0 1 0 0 0 → 0 ∗ 55 + 1 ∗ 21 + 0 ∗ 8 + 0 ∗ 3 + 0 ∗ 1 = 21
Binary numeration system 1 0 1 1 0 → 1 ∗ 2 4 + 0 ∗ 2 3 + 1 ∗ 2 2 + 1 ∗ 2 1 + 0 ∗ 2 0 = 22 Ternary numeration system (unique representation) 1 0 2 1 1 → 1 ∗ 3 4 + 0 ∗ 3 3 + 2 ∗ 3 2 + 1 ∗ 3 1 + 1 ∗ 3 0 = 138 A Fraenkel numeration system: ternary, but ... (unique representation) 1 0 2 1 1 → 1 ∗ 55 + 0 ∗ 21 + 2 ∗ 8 + 1 ∗ 3 + 1 ∗ 1 = 75 0 0 2 1 2 → 0 ∗ 55 + 0 ∗ 21 + 2 ∗ 8 + 1 ∗ 3 + 2 ∗ 1 = 21 0 1 0 0 0 → 0 ∗ 55 + 1 ∗ 21 + 0 ∗ 8 + 0 ∗ 3 + 0 ∗ 1 = 21 but unique representation by ternary strings with no 21*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 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 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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