A290134 -id:A290134 - cp's OEIS frontend

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

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

A290133 Number of unique X-rays of n X n binary matrices with exactly n ones.

Original entry on oeis.org

1, 1, 2, 5, 8, 13, 21, 31, 45, 65, 92, 127, 175, 237, 318, 425, 561, 735, 959, 1241, 1597, 2047, 2607, 3305, 4174, 5247, 6569, 8197, 10189, 12621, 15588, 19189, 23551, 28829, 35189, 42841, 52033, 63039, 76197, 91903, 110603, 132831, 159215, 190463, 227416

Offset: 0

Alois P. Heinz, Jul 20 2017

nonn

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

A unique X-ray allows reconstruction of the binary matrix.

			a(3) = 5: 00021, 00300, 02001, 10020, 12000.
a(4) = 8: 0000301, 0004000, 0030001, 0200020, 1000021, 1000300, 1030000, 1200001.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
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. A290052, A290134.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
      `if`(n=m or n=0, 1, add(b(n-i*j, min(n-i*j, i-1))*
      `if`(j=1, 2, 1), j=0..min(2, n/i)))))(i*(i+1))
    end:
a:= n-> `if`(n=0, 1, 1+b(n, n-1)) :
seq(a(n), n=0..60);

b[n_, i_] := b[n, i] = Function [m, If[n > m, 0, If[n == m || n == 0, 1, Sum[b[n - i*j, Min[n - i*j, i - 1]]*If[j == 1, 2, 1], {j, 0, Min[2, n/i]} ]]]][i*(i + 1)];
a[n_] := If[n == 0, 1, 1 + b[n, n - 1]] ;
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, May 06 2018

cp's OEIS Frontend

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

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