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A324224
Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.
4
1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 6, 18, 24, 18, 6, 1, 1, 8, 36, 96, 120, 96, 36, 8, 1, 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1, 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 35280, 15120, 4200, 840, 126, 14, 1
OFFSET
1,3
LINKS
Wikipedia, Permutation
FORMULA
T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)!/t! if t < n with t = |k|, T(n,k) = 0 otherwise.
T(n,k) = 1/|k|! * A324225(n,k).
E.g.f. of column k: x^t/t! * hypergeom([2, t], [t+1], x) with t = |k|+1.
Sum_{k=1-n..n-1} T(n,k) = A306495(n-1).
EXAMPLE
Triangle T(n,k) begins:
: 1 ;
: 1, 2, 1 ;
: 1, 4, 6, 4, 1 ;
: 1, 6, 18, 24, 18, 6, 1 ;
: 1, 8, 36, 96, 120, 96, 36, 8, 1 ;
: 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1 ;
: 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1 ;
MAPLE
b:= proc(s, c) option remember; (n-> `if`(n=0, c,
add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t<n, (n-t)*(n-1)!/t!, 0))(abs(k)):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
MATHEMATICA
T[n_, k_] := With[{t = Abs[k]}, If[t<n, (n-t)(n-1)!/t!, 0]];
Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 25 2021, after 3rd Maple program *)
CROSSREFS
Columns k=0-6 give (offsets may differ): A000142, A001563, A001286, A005990, A061206, A062199, A062148.
Row sums give A306495(n-1).
Cf. A132159 (right part of triangle), A306234, A324225.
Sequence in context: A327639 A273891 A034870 * A373432 A264622 A275017
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Feb 18 2019
STATUS
approved