A290057 -id:A290057 - cp's OEIS frontend

A290052 Number of X-rays of n X n binary matrices with exactly n ones.

1, 1, 4, 23, 139, 860, 5393, 34142, 217717, 1396346, 8997695, 58205686, 377775385, 2458841504, 16043226825, 104901986083, 687221188145, 4509605878736, 29636894936761, 195035340954186, 1285062484293880, 8476508261617168, 55969236979211755, 369900194873712830

Offset: 0

Alois P. Heinz, Jul 19 2017

nonn

The X-ray of a matrix is defined as the sequence of antidiagonal sums.

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
Index entries for sequences related to binary matrices

Main diagonal of A290057.

Cf. A019589, A014062, A290133.

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 *)

A019589(n) <= a(n) <= A014062(n).

a(n) ~ c * 3^(3*n) / (2^(2*n) * sqrt(n)), where c = 0.153294749730773567280925277269616968259180871352428154276351832424636097919... - Vaclav Kotesovec, Jul 22 2017

A290058 Number of X-rays of n X n binary matrices with exactly floor(n^2/2) ones.

Original entry on oeis.org

1, 1, 4, 30, 440, 9892, 331950, 15121926, 915201732, 70120569074, 6696969703276, 774618119733020, 107284227278413622, 17455779156567652806, 3307802489634916900474, 720231707823173636419042, 178973636259170839478327332, 50249140887232774758578932120

Offset: 0

Alois P. Heinz, Jul 19 2017

nonn

The X-ray of a matrix is defined as the sequence of antidiagonal sums.

			a(2) = 4: 011, 020, 101, 110.
a(3) = 30: 00121, 00211, 00220, 00301, 00310, 01021, 01111, 01120, 01201, 01210, 01300, 02011, 02020, 02101, 02110, 02200, 10021, 10111, 10120, 10201, 10210, 10300, 11011, 11020, 11101, 11110, 11200, 12001, 12010, 12100.

Alois P. Heinz, Table of n, a(n) for n = 0..253
C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, On the X-rays of permutations, arXiv:math/0506334 [math.CO], 2005.
Index entries for sequences related to binary matrices

Cf. A290057, A290134.

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(iquo(n^2, 2), n, 1):
seq(a(n), n=0..20);

b[n_, i_, t_]:= b[n, i, t] = Function[{m, jm}, If[n>m, 0, If[n==m, 1, Sum[b[n-j, i-t, 1-t], {j, 0, jm}]]]][i*(i+1-t), Min[i, n]]; a[n_]:=  b[Quotient[n^2, 2], n, 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Aug 09 2017, translated from Maple *)

a(n) = A290057(n,floor(n^2/2)).

a(n) ~ 6*sqrt(Pi) * n^(2*n+1/2) / exp(2*n). - Vaclav Kotesovec, Jul 22 2017

cp's OEIS Frontend

A290052 Number of X-rays of n X n binary matrices with exactly n ones.

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