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• Touchard Polynomials, Stirling Numbers and Random Permutations

Ross Pinsky

Department of Mathematics, Technion 32000 Haifa, ISRAEL

September 3, 2018

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Notation:

[n] = {1, 2, · · · , n}

Sn = permutations of [n]

|Sn| = n!

S6 3 σ1 = (

1 2 3 4 5 6 2 4 6 1 5 3

) = (124)(36)(5),

S6 3 σ2 = (

1 2 3 4 5 6 3 4 6 5 1 2

) = (136245) = (624513)

σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6.

There are (n − 1)! different n-cycles in Sn.

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Notation:

[n] = {1, 2, · · · , n}

Sn = permutations of [n]

|Sn| = n!

S6 3 σ1 = (

1 2 3 4 5 6 2 4 6 1 5 3

) = (124)(36)(5),

S6 3 σ2 = (

1 2 3 4 5 6 3 4 6 5 1 2

) = (136245) = (624513)

σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6.

There are (n − 1)! different n-cycles in Sn.

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Notation:

[n] = {1, 2, · · · , n}

Sn = permutations of [n]

|Sn| = n!

S6 3 σ1 = (

1 2 3 4 5 6 2 4 6 1 5 3

) = (124)(36)(5),

S6 3 σ2 = (

1 2 3 4 5 6 3 4 6 5 1 2

) = (136245) = (624513)

σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6.

There are (n − 1)! different n-cycles in Sn.

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Notation:

[n] = {1, 2, · · · , n}

Sn = permutations of [n]

|Sn| = n!

S6 3 σ1 = (

1 2 3 4 5 6 2 4 6 1 5 3

) = (124)(36)(5),

S6 3 σ2 = (

1 2 3 4 5 6 3 4 6 5 1 2

) = (136245) = (624513)

σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6.

There are (n − 1)! different n-cycles in Sn.

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Notation:

[n] = {1, 2, · · · , n}

Sn = permutations of [n]

|Sn| = n!

S6 3 σ1 = (

1 2 3 4 5 6 2 4 6 1 5 3

) = (124)(36)(5),

S6 3 σ2 = (

1 2 3 4 5 6 3 4 6 5 1 2

) = (136245) = (624513)

σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6.

There are (n − 1)! different n-cycles in Sn.

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Notation:

[n] = {1, 2, · · · , n}

Sn = permutations of [n]

|Sn| = n!

S6 3 σ1 = (

1 2 3 4 5 6 2 4 6 1 5 3

) = (124)(36)(5),

S6 3 σ2 = (

1 2 3 4 5 6 3 4 6 5 1 2

) = (136245) = (624513)

σ2 is a 6-cycle in S6. There are 5! different 6-cycles in S6.

There are (n − 1)! different n-cycles in Sn.

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Rising Factorials

x (n) := x(x +1) · · · (x +n−1)

( ≡

n∑ k=1

ankx k ) , x ∈ R, n ≥ 1. (1)

Unsigned Stirling Numbers of the First Kind: |s(n, k)| := the number of permutations in Sn with exactly k cycles, 1 ≤ k ≤ n Recurrence relation

|s(n, n)| = 1, |s(n, 1)| = (n − 1)!, n ≥ 1; |s(n + 1, k)| = n|s(n, k)|+ |s(n, k − 1)|, 2 ≤ k ≤ n.

(2)

It is easy to check that the coefficients {ank} in (1) also satisfy (2); thus, ank = |s(n, k)|. Therefore

x (n) = n∑

k=1

|s(n, k)|xk . (3)

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Rising Factorials

x (n) := x(x +1) · · · (x +n−1) ( ≡

n∑ k=1

ankx k ) , x ∈ R, n ≥ 1. (1)

Unsigned Stirling Numbers of the First Kind: |s(n, k)| := the number of permutations in Sn with exactly k cycles, 1 ≤ k ≤ n Recurrence relation

|s(n, n)| = 1, |s(n, 1)| = (n − 1)!, n ≥ 1; |s(n + 1, k)| = n|s(n, k)|+ |s(n, k − 1)|, 2 ≤ k ≤ n.

(2)

It is easy to check that the coefficients {ank} in (1) also satisfy (2); thus, ank = |s(n, k)|. Therefore

x (n) = n∑

k=1

|s(n, k)|xk . (3)

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Rising Factorials

x (n) := x(x +1) · · · (x +n−1) ( ≡

n∑ k=1

ankx k ) , x ∈ R, n ≥ 1. (1)

Unsigned Stirling Numbers of the First Kind: |s(n, k)| := the number of permutations in Sn with exactly k cycles, 1 ≤ k ≤ n

Recurrence relation

|s(n, n)| = 1, |s(n, 1)| = (n − 1)!, n ≥ 1; |s(n + 1, k)| = n|s(n, k)|+ |s(n, k − 1)|, 2 ≤ k ≤ n.

(2)

It is easy to check that the coefficients {ank} in (1) also satisfy (2); thus, ank = |s(n, k)|. Therefore

x (n) = n∑

k=1

|s(n, k)|xk . (3)

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Rising Factorials

x (n) := x(x +1) · · · (x +n−1) ( ≡

n∑ k=1

ankx k ) , x ∈ R, n ≥ 1. (1)

Unsigned Stirling Numbers of the First Kind: |s(n, k)| := the number of permutations in Sn with exactly k cycles, 1 ≤ k ≤ n Recurrence relation

|s(n, n)| = 1, |s(n, 1)| = (n − 1)!, n ≥ 1; |s(n + 1, k)| = n|s(n, k)|+ |s(n, k − 1)|, 2 ≤ k ≤ n.

(2)

It is easy to check that the coefficients {ank} in (1) also satisfy (2); thus, ank = |s(n, k)|. Therefore

x (n) = n∑

k=1

|s(n, k)|xk . (3)

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Rising Factorials

x (n) := x(x +1) · · · (x +n−1) ( ≡

n∑ k=1

ankx k ) , x ∈ R, n ≥ 1. (1)

Unsigned Stirling Numbers of the First Kind: |s(n, k)| := the number of permutations in Sn with exactly k cycles, 1 ≤ k ≤ n Recurrence relation

|s(n, n)| = 1, |s(n, 1)| = (n − 1)!, n ≥ 1; |s(n + 1, k)| = n|s(n, k)|+ |s(n, k − 1)|, 2 ≤ k ≤ n.

(2)

It is easy to check that the coefficients {ank} in (1) also satisfy (2); thus, ank = |s(n, k)|.

Therefore

x (n) = n∑

k=1

|s(n, k)|xk . (3)

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Rising Factorials

x (n) := x(x +1) · · · (x +n−1) ( ≡

n∑ k=1

ankx k ) , x ∈ R, n ≥ 1. (1)

Unsigned Stirling Numbers of the First Kind: |s(n, k)| := the number of permutations in Sn with exactly k cycles, 1 ≤ k ≤ n Recurrence relation

|s(n, n)| = 1, |s(n, 1)| = (n − 1)!, n ≥ 1; |s(n + 1, k)| = n|s(n, k)|+ |s(n, k − 1)|, 2 ≤ k ≤ n.

(2)

It is easy to check that the coefficients {ank} in (1) also satisfy (2); thus, ank = |s(n, k)|. Therefore

x (n) = n∑

k=1

|s(n, k)|xk . (3)

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Falling Factorials

(x)n := x(x − 1) · · · (x − n + 1), x ∈ R, n ≥ 1. (4)

Substituting −x for x in (4), we have (−x)n = −x(−x−1) · · · (−x−n+1) = (−1)nx(x+1) · · · (x+n−1) = (−1)nx (n). Substituting above −x for x , we have

(x)n = (−1)n(−x)(n) from (3)

= (−1)n n∑

k=1

|s(n, k)|(−x)k =

n∑ k=1

(−1)n−k |s(n, k)|xk . (5)

Define s(n, k) := (−1)n−k |s(n, k)|. The s(n, k) are called the Stirling Numbers of the First Kind. From (5) we have

(x)n = n∑

k=1

s(n, k)xk . (6)

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Falling Factorials

(x)n := x(x − 1) · · · (x − n + 1), x ∈ R, n ≥ 1. (4) Substituting −x for x in (4), we have (−x)n = −x(−x−1) · · · (−x−n+1) =

(−1)nx(x+1) · · · (x+n−1) = (−1)nx (n). Substituting above −x for x , we have

(x)n = (−1)n(−x)(n) from (3)

= (−1)n n∑

k=1

|s(n, k)|(−x)k =

n∑ k=1

(−1)n−k |s(n, k)|xk . (5)

Define s(n, k) := (−1)n−k |s(n, k)|. The s(n, k) are called the Stirling Numbers of the First Kind. From (5) we have

(x)n = n∑

k=1

s(n, k)xk . (6)

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Falling Factorials

(x)n := x(x − 1) · · · (x − n + 1), x ∈ R, n ≥ 1. (4) Substituting −x for x in (4), we have (−x)n = −x(−x−1) · · · (−x−n+1) = (−1)nx(x+1) · · · (x+n−1)

= (−1)nx (n). Substituting above −x for x , we have

(x)n = (−1)n(−x)(n) from (3)

= (−1)n n∑

k=1

|s(n, k)|(−x)k =

n∑ k=1

(−1)n−k |s(n, k)|xk . (5)

Define s(n, k) := (−1)n−k |s(n, k)|. The s(n, k) are called the Stirling Numbers of the First Kind. From (5) we have

(x)n = n∑

k=1

s(n, k)xk . (6)

Ross Pinsky Touchard Polynomials, Stirling Numbers and Random Permutations

• Falling Factorials

(x)n := x(x − 1) · · · (x − n + 1), x ∈ R, n ≥ 1. (4) Substituting −x for x in (4), we have (−x)n = −x(−x−1) · · · (−x−n+1) = (−1)nx(x+1) · · · (x+n−1) = (−1)nx (n).

Substituting above −x for x , we have

(x)n = (−1)n(−x)(n) from (3)

= (−1)n n∑

k=1

|s