A357329 - OEIS

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

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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 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!}]