Euler's partition theorem and the combinatorics of -sequences
Euler's partition theorem and the combinatorics of -sequences Euler's partition theorem and the combinatorics of -sequences
l ≥ 2 The l-Euler theorem [BME2]: The number of partitions of an integer N into parts from the set {a (l) 0 + a (l) 1 , a(l) 1 + a (l) 2 , a(l) 2 + a (l) 3 , . . .} is the same as the number of partitions of N in which the ratio of consecutive parts is greater than Proof: via lecture hall partitions. c l = l + √ l 2 − 4 2
l = 2 The l-Euler theorem [BME2]: The number of partitions of an integer N into parts from the set {0 + 1, 1 + 2, 2 + 3, . . .} = {1, 3, 5, . . .} is the same as the number of partitions of N in which the ratio of consecutive parts is greater than c 2 = 2 + √ 2 2 − 4 2 = 1
- 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: l-sequences For integer l ≥ 2, de
- 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 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
l = 2<br />
The l-Euler <strong><strong>the</strong>orem</strong> [BME2]: The number <strong>of</strong> <strong>partition</strong>s <strong>of</strong> an<br />
integer N into parts from <strong>the</strong> set<br />
{0 + 1, 1 + 2, 2 + 3, . . .} = {1, 3, 5, . . .}<br />
is <strong>the</strong> same as <strong>the</strong> number <strong>of</strong> <strong>partition</strong>s <strong>of</strong> N in which <strong>the</strong> ratio <strong>of</strong><br />
consecutive parts is greater than<br />
c 2 = 2 + √ 2 2 − 4<br />
2<br />
= 1
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