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A306455

Total number of covered falling diagonals in all n X n permutation matrices.

0, 1, 3, 14, 73, 454, 3253, 26480, 241505, 2440538, 27075301, 327197452, 4278799105, 60205974230, 907025841317, 14567520651224, 248474458923073, 4485765986251570, 85454391074596165, 1713134893536617348, 36052727133118151201, 794697884305583064302

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OFFSET

0,3

COMMENTS

A covered diagonal in a permutation matrix contains at least one 1.

Alternatively: Total number of covered raising diagonals in all n X n permutation matrices.

Also one half of the total number of all covered diagonals in all n X n permutation matrices.

Sum over all permutations p of [n] of the cardinality of the (signed) displacement set {p(i)-i, i=1..n}.

Alternatively: Sum over all permutations p of [n] of the cardinality of the set {p(i)+i, i=1..n}.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..450

Wikipedia, Permutation

Wikipedia, Permutation matrix

FORMULA

E.g.f.: (exp(-x)*(x+1)+x-1)/(x-1)^2.

a(n) = ((2*n^2-5*n+1)*a(n-1) - (n-1)*(n^2-4*n+2)*a(n-2) - (n-2)*(n-1)^2*a(n-3)) / (n-2) for n > 2, a(n) = n*(n+1)/2 for n < 3.

a(n) = Sum_{k=1..n} k * A125182(n,k).

a(n) = A259834(n+2) - n!.

a(n) = Sum_{k=1-n..n-1} A306461(n,k).

a(n) = Sum_{k=1-n..n-1} |k|! * A306234(n,k).

a(n) mod 2 = 1 - (n mod 2) = A059841(n) for n >= 2.

EXAMPLE

The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement sets {p(i)-i, i=1..3}: {0}, {-1,0,1}, {-1,0,1}, {-2,1}, {-1,2}, {-2,0,2}, representing the indices of covered falling diagonals in the permutation matrices

[1 ] [1 ] [ 1 ] [ 1 ] [ 1] [ 1]

[ 1 ] [ 1] [1 ] [ 1] [1 ] [ 1 ]

[ 1] [ 1 ] [ 1] [1 ] [ 1 ] [1 ] , respectively, the sum of the set cardinalities gives a(3) = 1 + 3 + 3 + 2 + 2 + 3 = 14.

MAPLE

a:= proc(n) option remember; `if`(n<3, n*(n+1)/2,

((2*n^2-5*n+1)*a(n-1)-(n-1)*(n^2-4*n+2)*a(n-2)

-(n-2)*(n-1)^2*a(n-3))/(n-2))

end:

seq(a(n), n=0..23);

MATHEMATICA

a[n_] := a[n] = If[n<3, n(n+1)/2, ((2n^2-5n+1) a[n-1] -

(n-1)(n^2-4n+2) a[n-2] - (n-2)(n-1)^2 a[n-3])/(n-2)];