login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A324225 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows. 3
1, 1, 2, 1, 2, 4, 6, 4, 2, 6, 12, 18, 24, 18, 12, 6, 24, 48, 72, 96, 120, 96, 72, 48, 24, 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120, 720, 1440, 2160, 2880, 3600, 4320, 5040, 4320, 3600, 2880, 2160, 1440, 720, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 35280, 30240, 25200, 20160, 15120, 10080, 5040 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,k) is the number of occurrences of k in all (signed) displacement lists [p(i)-i, i=1..n] of permutations p of [n].

LINKS

Alois P. Heinz, Rows n = 1..100, flattened

Wikipedia, Permutation

Wikipedia, Permutation matrix

FORMULA

T(n,k) = T(n,-k).

T(n,k) = (n-t)*(n-1)! if t < n with t = |k|, T(n,k) = 0 otherwise.

T(n,k) = |k|! * A324224(n,k).

E.g.f. of column k: x^t/t * hypergeom([2, t], [t+1], x) with t = |k|+1.

|T(n,k)-T(n,k-1)| = (n-1)! for k = 1-n..n.

EXAMPLE

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

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

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

  [    1]  [  1  ]  [    1]  [1    ]  [  1  ]  [1    ] , respectively. Indices -2 and 2 occur twice, -1 and 1 occur four times, and 0 occurs six times. So row n=3 is [2, 4, 6, 4, 2].

Triangle T(n,k) begins:

  :                             1                           ;

  :                        1,   2,   1                      ;

  :                   2,   4,   6,   4,   2                 ;

  :              6,  12,  18,  24,  18,  12,   6            ;

  :        24,  48,  72,  96, 120,  96,  72,  48,  24       ;

  :  120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120  ;

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), 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)!, 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)!, 0]];

Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 8}] // Flatten (* Jean-Fran├žois Alcover, Mar 25 2021, after 3rd Maple program *)

CROSSREFS

Columns k=0-6 give (offsets may differ): A000142, A001563, A062119, A052571, A052520, A282822, A052521.

Row sums give A001563.

T(n+1,n) gives A000142.

T(n+1,n-1) gives A052849.

T(n+1,n-2) gives A052560 for n>1.

Cf. A152883 (right half of this triangle without center column), A162608 (left half of this triangle), A306461, A324224.

Sequence in context: A139145 A201635 A179787 * A214739 A296159 A283334

Adjacent sequences:  A324222 A324223 A324224 * A324226 A324227 A324228

KEYWORD

nonn,look,tabf

AUTHOR

Alois P. Heinz, Feb 18 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 24 02:27 EDT 2021. Contains 345413 sequences. (Running on oeis4.)