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 = 3 The l-Euler theorem [BME2]: The number of partitions of an integer N into parts from the set {0 + 1, 1 + 3, 3 + 8, . . .} = {1, 4, 11, 29, . . .} is the same as the number of partitions of N in which the ratio of consecutive parts is greater than c 3 = 3 + √ 3 2 − 4 2 = (3 + √ 5)/2
l = 3 The l-Euler theorem [BME2]: The number of partitions of an integer N into parts from the set {0 + 1, 1 + 3, 3 + 8, . . .} = {1, 4, 11, 29, . . .} is the same as the number of partitions of N in which the ratio of consecutive parts is greater than c 3 = 3 + √ 3 2 − 4 2 = (3 + √ 5)/2 Stanley: bijection?
- 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: l = 2 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
- Page 64 and 65: Question: What is the generating fu
l = 3<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 + 3, 3 + 8, . . .} = {1, 4, 11, 29, . . .}<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 3 = 3 + √ 3 2 − 4<br />
2<br />
= (3 + √ 5)/2<br />
Stanley: bijection?
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