A345461 -id:A345461 - cp's OEIS frontend

A345464 Second column of A345461 - Number of distinct permutations after one step of the "optimist" sorting algorithm.

1, 1, 6, 38, 232, 1607, 12984, 117513, 1182540, 13060248, 157314056, 2051314949

Offset: 2

Olivier Gérard, Jun 20 2021

Cf. A345453, A345461.
Cf. A000166 (equivalent for Eulerian numbers).
Cf. A180191 (equivalent for Stirling numbers of the first kind).

a(11)-a(13) from Pontus von Brömssen, Jun 29 2021

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.

Original entry on oeis.org

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

Olivier Gérard, Jun 20 2021

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).

			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;
  ...

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

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

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.

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

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 *)

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)

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A345464 Second column of A345461 - Number of distinct permutations after one step of the "optimist" sorting algorithm.

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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.

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