Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:42
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Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:42 1.72 Prove the q-Vandermonde identity 72 (r-k) (...
Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:42 n- k n- k
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:42
n- k
n- k
Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:35 1.59 Determine the number of solutions of XI +··...
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:35
1.59 Determine the number of solutions of XI +···+ Xk integers; in non-negative integers. n in positive
Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:29 S(n) is the set of all permutations of {1, 2, ....
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:29
S(n) is the set of all permutations of {1, 2, . . .
,n}.
1.38 What is the expected number of fixed points when all n! permut:a tions of S(n) are equally likely?
Book: A Course in Enumeration. Author: Martin Aigner Chapter 2 Page:76 2.41 Derive the 2-term recurrence...
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 2 Page:76
2.41 Derive the 2-term recurrence for the Lah numbers and the Lah in- version formula
2.41 Derive the 2-term recurrence for the Lah numbers and the Lah in- version formula
Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:60 If A(z) = n≥0 , then an is the n-th coefficient...
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:60
If A(z) = n≥0
,
then an is the n-th coefficient of A(z), for
which we also write = []A(z).
Sometimes it is useful to consider
coefficients with negative index, with the understanding that
ak = 0 for k < 0. The coefficient a0 is called the constant
coefficient,
and we also write a0 = A(0). The set of all formal series
over
C shall be denoted by C[[z]]....
Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:35 1.60 Let e(n),o(n), and sc(n) denote, respective...
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:35
1.60 Let e(n),o(n), and sc(n) denote, respectively, the number of par- titions of n with an even number of even parts, with an odd number of even parts, and that are self-conjugate. Show that e(n) - o(n) - sc(n).
Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:29 According to this chapter, I think S n,k is the Stirling number and maybe the first kind. 1.37 Use the polynomial method to show...
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:29
According to this chapter, I think S n,k is the Stirling number
and maybe the first kind.
1.37 Use the polynomial method to show that sn lkti -o )sni Can you find a combinatorial proof?
1.37 Use the polynomial method to show that sn lkti -o )sni Can you find a combinatorial proof?
Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:29 1.41 Letl2. Show that the number of permutations...
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:29
1.41 Letl2. Show that the number of permutations a e S(n) that have a cycle of length l equals . What is the proportion t(n) of o e S(n) that contain a cycle of length > when all permutations are equally likely? Compute lim,- t(n)
1.41 Letl2. Show that the number of permutations a e S(n) that have a cycle of length l equals . What is the proportion...
Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:60 2.4 Let F(2)-「20 anz" with a0-0. Show that...
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:60
2.4 Let F(2)-「20 anz" with a0-0. Show that F(z) has a composi- tional inverse F(-1) (2)-Ση20Pnz" with bo 0 if and only if ai + 0. , 71
2.4 Let F(2)-「20 anz" with a0-0. Show that F(z) has a composi- tional inverse F(-1) (2)-Ση20Pnz" with bo 0 if and only if ai + 0. , 71
Book: A Course in Enumeration. Author: Martin Aigner Chapter 1 Page:9 1.10 Evaluate ΣΙ.1 12 and Σ.1 13 by counting configurations of dots as in the proof of Σ-is n(n+1) 1.10 Evaluate ΣΙ.1 12 and...
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:9
1.10 Evaluate ΣΙ.1 12 and Σ.1 13 by counting configurations of dots as in the proof of Σ-is n(n+1)
1.10 Evaluate ΣΙ.1 12 and Σ.1 13 by counting configurations of dots as in the proof of Σ-is n(n+1)
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:42
n- k
n- k
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:35
1.59 Determine the number of solutions of XI +···+ Xk integers; in non-negative integers. n in positive
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:29
S(n) is the set of all permutations of {1, 2, . . .
,n}.
1.38 What is the expected number of fixed points when all n! permut:a tions of S(n) are equally likely?
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 2 Page:76
2.41 Derive the 2-term recurrence for the Lah numbers and the Lah in- version formula
2.41 Derive the 2-term recurrence for the Lah numbers and the Lah in- version formula
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:60
If A(z) = n≥0
,
then an is the n-th coefficient of A(z), for
which we also write = []A(z).
Sometimes it is useful to consider
coefficients with negative index, with the understanding that
ak = 0 for k < 0. The coefficient a0 is called the constant
coefficient,
and we also write a0 = A(0). The set of all formal series
over
C shall be denoted by C[[z]]....
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:35
1.60 Let e(n),o(n), and sc(n) denote, respectively, the number of par- titions of n with an even number of even parts, with an odd number of even parts, and that are self-conjugate. Show that e(n) - o(n) - sc(n).
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:29
According to this chapter, I think S n,k is the Stirling number
and maybe the first kind.
1.37 Use the polynomial method to show that sn lkti -o )sni Can you find a combinatorial proof?
1.37 Use the polynomial method to show that sn lkti -o )sni Can you find a combinatorial proof?
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:29
1.41 Letl2. Show that the number of permutations a e S(n) that have a cycle of length l equals . What is the proportion t(n) of o e S(n) that contain a cycle of length > when all permutations are equally likely? Compute lim,- t(n)
1.41 Letl2. Show that the number of permutations a e S(n) that have a cycle of length l equals . What is the proportion...
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:60
2.4 Let F(2)-「20 anz" with a0-0. Show that F(z) has a composi- tional inverse F(-1) (2)-Ση20Pnz" with bo 0 if and only if ai + 0. , 71
2.4 Let F(2)-「20 anz" with a0-0. Show that F(z) has a composi- tional inverse F(-1) (2)-Ση20Pnz" with bo 0 if and only if ai + 0. , 71
Book: A Course in Enumeration. Author: Martin Aigner
Chapter 1 Page:9
1.10 Evaluate ΣΙ.1 12 and Σ.1 13 by counting configurations of dots as in the proof of Σ-is n(n+1)
1.10 Evaluate ΣΙ.1 12 and Σ.1 13 by counting configurations of dots as in the proof of Σ-is n(n+1)
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