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-sequences For integer l ≥ 2, define the sequence {a n (l) } n≥1 by a (l) n = la (l) n−1 − a(l) n−2 , with initial conditions a (l) 1 = 1, a (l) 2 = l. {a (3) n } = 1, 3, 8, 21, 55, 144, 377, . . . {a (2) n } = 1, 2, 3, 4, 5, 6, 7, . . .
l-sequences For integer l ≥ 2, define the sequence {a n (l) } n≥1 by a (l) n = la (l) n−1 − a(l) n−2 , with initial conditions a (l) 1 = 1, a (l) 2 = l. {a (3) n } = 1, 3, 8, 21, 55, 144, 377, . . . {a (2) n } = 1, 2, 3, 4, 5, 6, 7, . . . (These are the (k, l) sequences in [BME2] with k = l = l.)
- 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: 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 58 and 59: Define a q-analog of the l-nomial:
- Page 60 and 61: Another q-analog of the l-nomial Le
l-<strong>sequences</strong><br />
For integer l ≥ 2, define <strong>the</strong> sequence {a n<br />
(l) } n≥1 by<br />
a (l)<br />
n<br />
= la (l)<br />
n−1 − a(l) n−2 ,<br />
with initial conditions a (l)<br />
1<br />
= 1, a (l)<br />
2<br />
= l.<br />
{a (3)<br />
n } = 1, 3, 8, 21, 55, 144, 377, . . .<br />
{a (2)<br />
n } = 1, 2, 3, 4, 5, 6, 7, . . .<br />
(These are <strong>the</strong> (k, l) <strong>sequences</strong> in [BME2] with k = l = l.)
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