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

3,4

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

Links

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

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).

Example

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.

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 *)

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.

Sequence in context: A281385 A279270 A075464 * A247858 A247860 A196756

Adjacent sequences: A332706 A332707 A332708 * A332710 A332711 A332712

Keyword

nonn,tabl

Author

Peter Kagey, Feb 20 2020

Status

approved