A289971 Number of permutations of [n] determined by their antidiagonal sums.

1, 1, 2, 4, 9, 20, 49, 114, 277, 665, 1608, 3875

Offset: 0

Author

Martin Rubey, Jul 16 2017

Keywords

• nonn
• more

Links

• C. Bebeacua, T. Mansour, A. Postnikov, and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
• FindStat - Combinatorial Statistic Finder, The number of permutations with the same antidiagonal sums.
• Martin Rubey, Alternating Sign Matrices Through X-Rays, J. Int. Seq., Vol. 24 (2021), Article 21.6.5.

Crossrefs

Cf. A019589, A002047, A290279, A290280.

Programs

Mathematica

xray[perm_List] := Module[{P, n = Length[perm]}, P[, ] = 0; Thread[perm -> Range[n]] /. Rule[i_, j_] :> Set[P[i, j], 1]; Table[Sum[P[i - j + 1, j], {j, Max[1, i - n + 1], Min[i, n]}], {i, 1, 2n - 1}]];
a[n_] := xray /@ Permutations[Range[n]] // Tally // Count[#, {_List, 1}]&;
Do[Print[n, " ", a[n]], {n, 0, 10}] (* Jean-François Alcover, Feb 28 2020 *)

Sage

def X_ray(pi):
    P = Permutation(pi).to_matrix()
    n = P.nrows()
    return tuple(sum(P[k-1-j][j] for j in range(max(0, k-n), min(k,n)))
                 for k in range(1,2*n))
@cached_function
def X_rays(n):
    return sorted(X_ray(pi) for pi in Permutations(n))
def statistic(pi): return X_rays(pi.size()).count(X_ray(pi))
[[statistic(pi) for pi in Permutations(n)].count(1) for n in range(7)]

Extensions

a(8)-a(11) from Alois P. Heinz, Jul 24 2017