Euler's partition theorem and the combinatorics of -sequences

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Question: When l = 2, several refinements of Euler’s theorem follow from Sylvester’s bijection. What refinements of the l-Euler theorem can be obtained from the bijection? We have some partial answers, but here’s one we can’t answer: Sylvester showed that if, in his bijection, the partition µ into odd parts maps to λ (distinct parts), then the number of distinct part sizes occurring in µ is the same as the number of maximal chains in λ. (A chain is a sequence of consecutive integers.) Is there an analog for l > 2?
Question: Can we enumerate either of these finite sets: ◮ The integer sequences λ 1 , . . . , λ n in which the ratio of consecutive parts is greater than c l and λ 1 < a (l) n+1 ? We have a combinatorial characterization: these can be viewed as fillings of a staircase of shape (n, n − 1, . . . , 1) such that (i) the filling of each row is an l-ary string with no (l − 1)(l − 2) ∗ (l − 1) and (ii) the rows are weakly decreasing, lexicographically. For l = 2 the answer is 2 n . ◮ The integer sequences λ 1 , . . . , λ n satisfying 1 ≥ λ 1 a n ≥ λ 2 a n−1 ≥ . . . ≥ λ n−1 a 2 ≥ λ n a 1 ≥ 0 For l = 2 this is the same set as above and the answer is 2 n .
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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 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 64 and 65: Question: What is the generating fu
Page 66 and 67: CanaDAM 2009 2nd Canadian Discrete
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Euler's partition theorem and the combinatorics of -sequences

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