A357329 Triangular array read by rows: T(n, k) = number of occurrences of 2k as a sum |1 - p(1)| + |2 - p(2)| + ... + |n - p(n)|, where (p(1), p(2), ..., p(n)) ranges through the permutations of (1,2,...,n), for n >= 1, 0 <= k <= n-1.

1, 1, 1, 1, 2, 3, 1, 3, 7, 9, 1, 4, 12, 24, 35, 1, 5, 18, 46, 93, 137, 1, 6, 25, 76, 187, 366, 591, 1, 7, 33, 115, 327, 765, 1523, 2553, 1, 8, 42, 164, 524, 1400, 3226, 6436, 11323, 1, 9, 52, 224, 790, 2350, 6072, 13768, 27821, 50461, 1, 10, 63, 296, 1138, 3708, 10538, 26480, 59673, 121626, 226787

Offset

1,5

Comments

In the Name, (1,2,...,n) can be replaced by any of its permutations. The first 10 row sums are the first 10 terms of A263898.

Links

Alois P. Heinz, Rows n = 1..141, flattened

Example

First 8 rows:
  1
  1     1
  1     2     3
  1     3     7     9
  1     4    12    24     35
  1     5    18    46     93     137
  1     6    25    76    187     366    591
  1     7    33   115    327     765   1523    2553
For n=3, write
  123   123   123   123   123   123
  123   132   213   231   312   312
  000   011   110   112   211   211,
where row 3 represents |1 - p(1)| + |2 - p(2)| + |3 - p(n)| for the 6 permutations (p(1), p(2), p(2)) in row 3. The sums in row 3 are 0,2,2,4,4,4, so that the numbers 0, 2, 4 occur with multiplicities 1, 2, 3, as in row 3 of the array.

Maple

g:= proc(h, n) local i, j; j:= irem(h, 2, 'i');
       1-`if`(h=n, 0, (i+1)*z*t^(i+j)/g(h+1, n))
    end:
T:= n-> (p-> seq(coeff(p, t, k), k=0..n-1))
        (coeff(series(1/g(0, n), z, n+1), z, n)):
seq(T(n), n=1..12);  # Alois P. Heinz, Oct 02 2022

Mathematica

p[n_] := p[n] = Permutations[Range[n]];
f[n_, k_] := f[n, k] = Abs[p[n][[k]] - Range[n]]
c[n_, k_] := c[n, k] = Total[f[n, k]]
t[n_] := Table[c[n, k], {k, 1, n!}]
u = Table[Count[t[n], 2 m], {n, 1, 10}, {m, 0, n - 1}]  (* A357329, array *)
Flatten[u]  (* A357329, sequence *)

Crossrefs

Cf. A000142, A263898.

Subtriangle of A062869.

Sequence in context: A236918 A152821 A071943 * A062869 A102473 A011117

Adjacent sequences: A357326 A357327 A357328 * A357330 A357331 A357332

Keyword

nonn,tabl

Author

Clark Kimberling, Sep 24 2022

Status

approved