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
An l-nomial theorem [LS]: An analog of is n∑ k=0 ( n k) z k = (1 + z) n n∑ ( ) n (l) z k = k k=0 n−1 ∏ i=0 (u i−(n−1)/2 l + v i−(n−1)/2 l z) = (1 + ∆ (l) 1 z + z2 )(1 + ∆ (l) 3 z + z2 ) · · · (1 + ∆ (l) n−1 z + z2 ) (1 + z)(1 + ∆ (l) 2 z + z2 )(1 + ∆ (l) 4 z + z2 ) · · · (1 + ∆ (l) n−1 z + z2 ) n even n odd
A coin-flipping interpretation of the l-nomial For real r, let C r (l) be a weighted coin for which the probability of tails is ul r /∆(l) r and the probability of heads is vl r /∆(l) r . The probability of getting exactly k heads when tossing the n coins C (l) −(n−1)/2 , C (l) 1−(n−1)/2 , C (l) 2−(n−1)/2 , . . . , C (l) (n−1)/2 :
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- Page 33 and 34: The Lecture Hall Theorem [BME1] The
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- Page 50 and 51: Let u l and v l be the roots of the
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- Page 66 and 67: CanaDAM 2009 2nd Canadian Discrete
A coin-flipping interpretation <strong>of</strong> <strong>the</strong> l-nomial<br />
For real r, let C r<br />
(l) be a weighted coin for which <strong>the</strong> probability <strong>of</strong><br />
tails is ul r /∆(l) r <strong>and</strong> <strong>the</strong> probability <strong>of</strong> heads is vl r /∆(l) r .<br />
The probability <strong>of</strong> getting exactly k heads when tossing <strong>the</strong> n coins<br />
C (l)<br />
−(n−1)/2 ,<br />
C (l)<br />
1−(n−1)/2 ,<br />
C (l)<br />
2−(n−1)/2 , . . . , C (l)<br />
(n−1)/2 :
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