There are 6 pending statistics:

Identifier
Values
No modified entries

Description
The number of indices that are not small excedances.
A small excedance is an index $i$ for which $\pi_i \neq i+1$.
Diff Description
The number of indices that are not small excedances.

A small excedance is an index
i for which \pi_i \neq i+1.
Code
def statistic(pi):
return sum( Integer(1) for i in range(len(pi)) if pi[i] != i+2 )


Created
Feb 26, 2015 at 21:04 by Christian Stump
Updated
May 22, 2023 at 23:20 by Will Dowling

Martin Rubey
23 May 8:56
I think there is a typo ( eq should be =)

Comment:
Name:
E-Mail:
Please type the top row of Identifier
Values
No modified entries

Description
The number of indices that are not small weak excedances.
A small weak excedance is an index $i$ such that $\pi_i \notin \{i,i+1\}$.
Diff Description
The number of indices that are not small weak excedances.

A small weak excedance is an index
i such that \pi_i \notin \{i,i+1\}.
References
 Li, Y. Ménage Numbers and Ménage Permutations arXiv:1502.06068
Diff References
 Li, Y. Ménage Numbers and Ménage Permutations [[arXiv:1502.06068]]
Code
def statistic(pi):
return sum( 1 for i in range(len(pi)) if pi[i] not in [i+1,i+2]  )

Created
Feb 24, 2015 at 13:12 by Christian Stump
Updated
May 22, 2023 at 23:18 by Will Dowling

Martin Rubey
23 May 8:56
I think there is a typo ( eq should be =)
Christian Stump
23 May 9:05
"\ eq should be ="

Comment:
Name:
E-Mail:
Please type the top row of Identifier
Values
No modified entries

Description
The number of indices that are not cyclical small weak excedances.
A cyclical small weak excedance is an index $i$ such that $\pi_i \neq i+1$ considered cyclically.
Diff Description
The number of indices that are not cyclical small weak excedances.

A cyclical small weak excedance is an index
i such that \pi_i \neq i+1 considered cyclically.
References
 Li, Y. Ménage Numbers and Ménage Permutations arXiv:1502.06068
Diff References
 Li, Y. Ménage Numbers and Ménage Permutations [[arXiv:1502.06068]]
Code
def statistic(pi):
n = len(pi)
return sum( 1 for i in range(n) if pi[i] != ( (i+1) % n + 1 ) )


Created
Feb 24, 2015 at 13:16 by Christian Stump
Updated
May 22, 2023 at 23:17 by Will Dowling

Martin Rubey
23 May 8:57
I think there is a typo ( eq should be =)

Comment:
Name:
E-Mail:
Please type the top row of Identifier
Values
No modified entries

Description
The number of column-product over-primes.
This is the number of indices $i$ such that $i\pi(i)+1$ is prime.
Diff Description
The number of primes obtained by multiplying column-product over-preimage and image of a permutation and adding ones.

This is the number of indices i such that i\pi(i)+1 is prim
e.
References
 Sun, Z.-W. Primes arising from permutations MathOverflow:315259
Code
def statistic(pi):
return sum(1 for i in pi if is_prime(i*pi(i) + 1))

Created
Nov 14, 2018 at 09:02 by Martin Rubey
Updated
May 22, 2023 at 22:56 by Will Dowling

Martin Rubey
23 May 8:59
This is hard to read. Also, the first line should always contain a hint about which collection the statistic is in, eg. "of a permutation". Why "column"?

Comment:
Name:
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Please type the top row of Identifier
Values
No modified entries

Description
The number of column-product almost-primes.
This is the number of indices $i$ such that $i\pi(i)-1$ is prime. Note that the numbers $0$ and $1$ may arise, but are not prime.
Diff Description
The number of primes obtained by multiplying column-product almost-preimage and image of a permutation and subtracting one.

es.

This is the number of indices i such that i\pi(i)-1 is prime.
Note that the numbers 0 and 1 may arise, but are not prime.
References
 Sun, Z.-W. Primes arising from permutations MathOverflow:315259
Code
def statistic(pi):
return sum(1 for i in pi if is_prime(i*pi(i) - 1))

Created
Nov 14, 2018 at 09:09 by Martin Rubey
Updated
May 22, 2023 at 22:55 by Will Dowling

Martin Rubey
23 May 8:59
This is hard to read. Also, the first line should always contain a hint about which collection the statistic is in, eg. "of a permutation". Why "column"?

Comment:
Name:
E-Mail:
Please type the top row of Identifier
Values
No modified entries

Description
The number of column-sum primes.
This is the number of sums $i+\pi(i)$ that are prime.
Diff Description
The number of primecolumn-sum primes.

Thi
s ins the column sums of the two line notation of a permutationnumber of sums i+\pi(i) that are prime.
References
 Bradley, P. Prime Number Sums arXiv:1809.01012
Code
def statistic(pi):
return sum(1 for i in pi if is_prime(pi(i) + i))


Created
Nov 14, 2018 at 07:25 by Martin Rubey
Updated
May 22, 2023 at 22:53 by Will Dowling 