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
Examples
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.
Links
- Alois P. Heinz, Rows n = 1..141, flattened
Programs
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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 *)
Comments