This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
The permutation sums for n=2 are (1,2) + (1,2) = (2,4); (1,2) + (2,1) = (2,1) + (1,2) = (3,3); (2,1) + (2,1) = (4,2); so a(2)=3.
with(linalg): A019448 := proc(n) local i, j, m; m := array(1..n, 1..n); for i from 1 to n do for j from 1 to n do m[i, j] := a[i+j] od od; nops([coeffs(det(m))]); end; # Jeffrey Shallit, Jun 08 2000 with(LinearAlgebra): f:=n->nops([coeffs(Determinant(Matrix(n, (i, j) -> a[i+j] )))]): [seq(f(n), n=0..10)]; # Vaclav Kotesovec, Mar 29 2019
f[n_] := Length@ Expand@ Det@ Table[t[i + j], {i, n}, {j, n}]; Do[ Print@ f@n, {n, 11}] (* Robert G. Wilson v, Sep 17 2006 *)
def a362967(n): return Partitions(n^2, length=n, inner=range(n,0,-1), outer=range(2*n-1,n-1,-1)).cardinality()
a(0) = 1: []. a(1) = 1: 1. a(2) = 4: 011, 020, 101, 110. a(3) = 23: 00021, 00111, 00120, 00201, 00210, 00300, 01011, 01020, 01101, 01110, 01200, 02001, 02010, 02100, 10011, 10020, 10101, 10110, 10200, 11001, 11010, 11100, 12000.
b:= proc(n, i, t) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
add(b(n-j, i-t, 1-t), j=0..min(i, n)))))(i*(i+1-t))
end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..30);
b[n_, i_, t_] := b[n, i, t] = Function[m, If[n > m, 0, If[n == m, 1, Sum[b[n - j, i - t, 1 - t], {j, 0, Min[i, n]}]]]][i*(i + 1 - t)];
a[n_] := b[n, n, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)
def a381244(n): return HyperplaneArrangements(QQ,tuple(f'x{i}' for i in range(n)))([[list(r),0] for p in Permutations(n) for r in Permutations([p[i]-i-1 for i in range(n)]) if vector(r)>0]).n_regions()
with(LinearAlgebra): f:=n->nops([coeffs(Permanent(Matrix(n, (i, j) -> a[abs(i-j)])))]): [seq(f(n), n=1..12)]; # Vaclav Kotesovec, Mar 29 2019
Permanent[m_List] := With[{v = Array[x, Length@m]}, Coefficient[Times @@ (m . v), Times @@ v]]; ; f[n_] := Length@Expand@Permanent@Table[t[Abs[i - j]], {i, n}, {j, n}]; Array[f, 11] (* Robert G. Wilson v *)
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 *)
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)]
For n = 3, there are A019589(3) = 5 difference vectors up to permutation of components: (-2, 0, 2), (-2, 1, 1), (-1, -1, 2), (-1, 0, 1), and (0, 0, 0). However, (-2, 0, 2) and (-1, 0, 1) are the same up to a factor 2, and (-2, 1, 1) and (-1, -1, 2) are the same up to negation and reversing the order. Hence, a(3) = 3.
For n = 3, there are A019589(3) = 5 difference vectors up to permutation of components: (-2, 0, 2), (-2, 1, 1), (-1, -1, 2), (-1, 0, 1), and (0, 0, 0). However, (-2, 0, 2) and (-1, 0, 1) are the same up to a factor 2. Hence, a(3) = 4.
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