A332709 Triangle T(n,k) read by rows where T(n,k) is the number of ménage permutations that map 1 to k, with 3 <= k <= n.

1, 1, 1, 4, 5, 4, 20, 20, 20, 20, 115, 116, 117, 116, 115, 787, 791, 791, 791, 791, 787, 6184, 6203, 6204, 6205, 6204, 6203, 6184, 54888, 55000, 55004, 55004, 55004, 55004, 55000, 54888, 542805, 543576, 543595, 543596, 543597, 543596, 543595, 543576, 542805

Offset: 3

Author

Peter Kagey, Feb 20 2020

Keywords

• nonn
• tabl

Comments

Rows are palindromic.

Conjecture: Rows are unimodal (i.e., increasing, then decreasing).

Conjecture: T(n,k) - T(n,k-1) = A127548(n-2k+4) for n >= 2k - 3. - Peter Kagey, Jan 22 2021

Examples

			Triangle begins:
  n\k|     3      4      5      6      7      8      9     10
  ---+--------------------------------------------------------
   3 |     1
   4 |     1,     1
   5 |     4,     5,     4
   6 |    20,    20,    20,    20
   7 |   115,   116,   117,   116,   115
   8 |   787,   791,   791,   791,   791,   787
   9 |  6184,  6203,  6204,  6205,  6204,  6203,  6184
  10 | 54888, 55000, 55004, 55004, 55004, 55004, 55000, 54888
For n=5 and k=3, the T(5,3)=4 permutations on five letters that start with a 3 are 31524, 34512, 35124, and 35212.
		

Links

• Peter Kagey, Table of n, a(n) for n = 3..1277 (first 50 rows)

Crossrefs

Cf. A127548.

First column given by A258664.

Second column given by A258665.

Third column given by A258666.

Fourth column given by A258667.

Row sums given by A000179.

Programs

Mathematica

T[n_, k_] :=
Sum[Sum[(-1)^i*(n - i - 1)!*Binomial[2*k - j - 4, j]*
    Binomial[2*(n - k + 1) - i + j, i - j], {j, Max[k + i - n - 1, 0],
     Min[i, k - 2]}], {i, 0, n - 1}]
(* Peter Kagey, Jan 22 2021 *)

Formula

T(n,k) = Sum_{i=0..n-1} Sum_{j=max(k+i-n-1,0)..min(i,k-2)} (-1)^i*(n-i-1)! * binomial(2k-j-4,j) * binomial(2(n-k+1)-i+j,i-j).