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
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.
Programs
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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
Comments