A345462 Triangle T(n,k) (n >= 1, 0 <= k <= n-1) read by rows: number of distinct permutations after k steps of the "first transposition" algorithm.

1, 2, 1, 6, 3, 1, 24, 13, 4, 1, 120, 67, 23, 5, 1, 720, 411, 146, 36, 6, 1, 5040, 2921, 1067, 272, 52, 7, 1, 40320, 23633, 8800, 2311, 456, 71, 8, 1, 362880, 214551, 81055, 21723, 4419, 709, 93, 9, 1, 3628800, 2160343, 825382, 224650, 46654, 7720, 1042, 118, 10, 1

Offset

1,2

Comments

The first transposition algorithm is: if the permutation is sorted, then exit; otherwise, exchange the first unsorted letter with the letter currently at its index. Repeat.

At each step at least 1 letter (possibly 2) is sorted.

If one counts the steps necessary to reach the identity, this gives the Stirling numbers of the first kind (reversed).

References

D. E. Knuth, The Art of Computer Programming, Vol. 3 / Sorting and Searching, Addison-Wesley, 1973.

Links

Alois P. Heinz, Rows n = 1..150, flattened

Formula

T(n,0) = n!; T(n,n-3) = (3*(n-1)^2 - n + 3)/2.

From Alois P. Heinz, Aug 11 2021: (Start)

T(n,k) = T(n,k-1) - A010027(n,n-k) for k >= 1.

T(n,k) - T(n,k+1) = A123513(n,k).

T(n,0) - T(n,1) = A000255(n-1) for n >= 2.

T(n,1) - T(n,2) = A000166(n) for n >= 3.

T(n,2) - T(n,3) = A000274(n) for n >= 4.

T(n,3) - T(n,4) = A000313(n) for n >= 5. (End)

Example

Triangle begins:
      1;
      2,     1;
      6,     3,    1;
     24,    13,    4,    1;
    120,    67,   23,    5,   1;
    720,   411,  146,   36,   6,  1;
   5040,  2921, 1067,  272,  52,  7, 1;
  40320, 23633, 8800, 2311, 456, 71, 8, 1;
  ...

Maple

b:= proc(n, k) option remember; (k+1)!*
      binomial(n, k)*add((-1)^i/i!, i=0..k+1)/n
    end:
T:= proc(n, k) option remember;
     `if`(k=0, n!, T(n, k-1)-b(n, n-k+1))
    end:
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Aug 11 2021

Mathematica

b[n_, k_] := b[n, k] = (k+1)!*Binomial[n, k]*Sum[(-1)^i/i!, {i, 0, k+1}]/n;
T[n_, k_] := T[n, k] = If[k == 0, n!, T[n, k-1] - b[n, n-k+1]];
Table[Table[T[n, k], {k, 0, n - 1}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)

Crossrefs

Cf. A321352, A345461 (same idea for other sorting algorithms).

Cf. A180191 (second column, k=1).

Cf. A107111 a triangle with some common parts.

Cf. A143689 (diagonal T(n,n-3)).

Cf. A000142, A000166, A000255, A000274, A000313, A010027, A123513.

Sequence in context: A173333 A221915 A249619 * A222159 A221623 A096334

Adjacent sequences: A345459 A345460 A345461 * A345463 A345464 A345465

Keyword

nonn,tabl

Author

Olivier Gérard, Jun 20 2021

Status

approved