cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A290818 Array read by antidiagonals: T(m,n) = number of irredundant sets in the lattice (rook) graph K_m X K_n.

Original entry on oeis.org

2, 3, 3, 4, 11, 4, 5, 24, 24, 5, 6, 47, 94, 47, 6, 7, 88, 272, 272, 88, 7, 8, 163, 774, 1185, 774, 163, 8, 9, 304, 2230, 4280, 4280, 2230, 304, 9, 10, 575, 6542, 15781, 20106, 15781, 6542, 575, 10, 11, 1104, 19452, 60604, 88512, 88512, 60604, 19452, 1104, 11
Offset: 1

Views

Author

Andrew Howroyd, Aug 11 2017

Keywords

Examples

			Array begins:
===============================================================
m\n| 1   2     3      4       5        6        7         8
---+-----------------------------------------------------------
1  | 2   3     4      5       6        7        8         9 ...
2  | 3  11    24     47      88      163      304       575 ...
3  | 4  24    94    272     774     2230     6542     19452 ...
4  | 5  47   272   1185    4280    15781    60604    240073 ...
5  | 6  88   774   4280   20106    88512   400728   1879744 ...
6  | 7 163  2230  15781   88512   453271  2326534  12363513 ...
7  | 8 304  6542  60604  400728  2326534 13169346  76446456 ...
8  | 9 575 19452 240073 1879744 12363513 76446456 476777153 ...
...

Links

Crossrefs

Row 2 is A290707 for n > 1.
Main diagonal is A290586.
Cf. A287274, A290632.

Programs

  • Mathematica
    s[n_, k_]:=Sum[(-1)^i*Binomial[n, i] StirlingS2[n - i, k - i], {i, 0, Min[n, k]}];
c[m_, n_, x_]:=Sum[Binomial[m, i] (n^i - n!*StirlingS2[i, n])*x^i, {i, 0, m - 1}];
p[m_, n_, x_]:=Sum[Sum[Binomial[m, k] Binomial[n, r]* k!*s[r, k]*x^r*c[m - k, n - r, x], {r, 2k, n - 1}], {k,0, m - 1}];
b[m_, n_, x_]:=m^n*x^n + n^m*x^m - If[n<=m, n!*x^m*StirlingS2[m, n], m!*x^n*StirlingS2[n, m]];
T[m_, n_]:= b[m, n, 1] + p[m, n, 1];
Table[T[n, m -n + 1], {m, 10}, {n, m}]//Flatten
(* Indranil Ghosh, Aug 12 2017, after PARI code *)
  • PARI
    \\ See A. Howroyd note in A290586 for explanation.
s(n,k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
c(m,n,x)=sum(i=0, m-1, binomial(m, i) * (n^i - n!*stirling(i, n, 2))*x^i);
p(m,n,x)={sum(k=0, m-1, sum(r=2*k, n-1, binomial(m, k) * binomial(n, r) * k! * s(r, k) * x^r * c(m-k, n-r, x) ))}
b(m,n,x) = m^n*x^n + n^m*x^m - if(n<=m, n!*x^m*stirling(m, n, 2), m!*x^n*stirling(n, m, 2));
T(m,n) = b(m,n,1) + p(m,n,1);
for(m=1,10,for(n=1,m,print1(T(n,m-n+1),", ")));

Formula

T(m,n) = A290632(m, n) + Sum_{k=0..m-1} Sum_{r=2*k..n-1} binomial(m,k) * binomial(n,r) * k! * A008299(r,k) * c(m-k,n-r) where c(m,n) = Sum_{i=0..m-1} binomial(n,i) * (n^i - n!*stirling2(i, n)).

A290823 Irregular triangle read by rows: T(n,k) = number of k-irredundant sets in the n X n rook graph.

Original entry on oeis.org

1, 1, 1, 1, 4, 6, 1, 9, 36, 48, 1, 16, 120, 416, 632, 1, 25, 300, 1900, 6550, 10930, 400, 1, 36, 630, 6240, 37080, 128592, 240192, 39600, 900, 1, 49, 1176, 16660, 149695, 858774, 3064656, 6354866, 2492385, 229320, 1764
Offset: 0

Views

Author

Andrew Howroyd, Aug 11 2017

Keywords

Comments

For each row, k lies in the range 0..max(n, 2*n-4). The upper limit is the upper irredundance number of the graph.

Examples

			Triangle begins:
1;
1,  1;
1,  4,   6;
1,  9,  36,   48;
1, 16, 120,  416,   632;
1, 25, 300, 1900,  6550,  10930,    400;
1, 36, 630, 6240, 37080, 128592, 240192, 39600, 900;
...
As polynomials these are 1; 1 + x; 1 + 4*x + 6*x^2; etc.

Links

Crossrefs

Row sums are A290586.

Programs

  • Mathematica
    s[n_, k_]:=Sum[(-1)^i*Binomial[n, i] StirlingS2[n - i, k - i], {i, 0, Min[n, k]}]; c[m_, n_, x_]:=Sum[Binomial[m, i] (n^i - n !*StirlingS2[i, n])*x^i, {i, 0, m - 1}]; p[m_, n_, x_]:=Sum[Sum[Binomial[m, k] Binomial[n, r]* k!*s[r, k]*x^r*c[m - k, n - r, x], {r, 2k, n - 1}], {k,0, m - 1}]; a[n_, x_]:=(2*n^n - n !)x^n + p[n, n, x]; A[n_]:=If[n==0, {1},  Drop[Block[{q=a[n, x]}, CoefficientList[q + x^(Exponent[q, x] + 1), x]], -1]]; Table[A[n], {n, 0, 15}] (* Indranil Ghosh, Aug 12 2017, after PARI code *)
  • PARI
    \\ see A. Howroyd note in A290586 for explanation
s(n,k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
c(m,n,x)=sum(i=0, m-1, binomial(m, i) * (n^i - n!*stirling(i, n, 2))*x^i);
p(m, n, x)={sum(k=0, m-1, sum(r=2*k, n-1, binomial(m, k) * binomial(n, r) * k! * s(r, k) * x^r * c(m-k, n-r, x) ))}
a(n,x) = (2*n^n - n!)*x^n + p(n,n,x);
for (n=0,8,my(q=a(n,x));print(Vec(q+O(x^(poldegree(q)+1)) )))

Formula

T(n, 0) = 1.
T(n, 1) = n^2.
T(n, 2) = binomial(n^2, 2).
T(n, 3) = binomial(n^2, 3) - n^2*(n-1)^2.
T(n, 2*n-4) = n^2*(n-1)^2 for n > 4.

A291104 Number of maximal irredundant sets in the n X n rook graph.

Original entry on oeis.org

1, 6, 48, 632, 10130, 194292, 4730810, 145114944, 5529662802, 256094790500, 14038667879522, 890349688082736, 64160617557387338, 5183023418382933060, 464623151635449639450, 45857185726197195813632, 4951604249874284663582498, 581839639424819461006405956
Offset: 1

Views

Author

Eric W. Weisstein, Aug 17 2017

Keywords

Links

Crossrefs

Main diagonal of A291543.
Cf. A008299, A248744, A290586.

Programs

  • Mathematica
    (* Start *)
s[n_, k_] := Sum[(-1)^i Binomial[n, i] StirlingS2[n - i, k - i], {i, 0, Min[n, k]}]
p[m_, n_, x_] := Sum[Binomial[m, k] Binomial[n, j] k! s[n - j, k - 1] j! StirlingS2[m - k, j - 1] x^(m + n - j - k), {k, 2, m - 2}, {j, 2, m - k}]
a[n_] := 2 n^n - n! + p[n, n, 1]
Array[a, 20]
(* End *)
  • PARI
    \\ here s(n, k) is A008299, 2*n^n - n! is A248744.
s(n, k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
p(m, n, x)={sum(k=2, m-2, sum(j=2, m-k, binomial(m, k) * binomial(n, j) * k! * s(n-j, k-1) * j! * stirling(m-k, j-1, 2) * x^(m+n-j-k) ))}
a(n) = 2*n^n - n! + p(n,n,1); \\ Andrew Howroyd, Aug 25 2017

Formula

a(n) = 2*n^n - n! + Sum_{k=2..n-2} Sum_{j=2..n-k} binomial(n,k) * binomial(n,j) * k! * A008299(n-j,k-1) * j! * stirling2(n-k,j-1). - Andrew Howroyd, Aug 25 2017

Extensions

Terms a(5) and beyond from Andrew Howroyd, Aug 25 2017

A291543 Array read by antidiagonals: T(m,n) = number of maximal irredundant sets in the lattice (rook) graph K_m X K_n.

Original entry on oeis.org

1, 2, 2, 3, 6, 3, 4, 11, 11, 4, 5, 18, 48, 18, 5, 6, 27, 109, 109, 27, 6, 7, 38, 218, 632, 218, 38, 7, 8, 51, 405, 1649, 1649, 405, 51, 8, 9, 66, 724, 4192, 10130, 4192, 724, 66, 9, 10, 83, 1277, 10889, 34801, 34801, 10889, 1277, 83, 10
Offset: 1

Views

Author

Andrew Howroyd, Aug 25 2017

Keywords

Comments

Maximal irredundant sets can be either dominating or not. The dominating maximal irredundant sets are the minimal dominating sets (A290632). The non-dominating maximal irredundant sets are those irredundant sets that have exactly one empty row and one empty column and at least one row and one column with more than one element. See note in A290586 for some additional details.

Examples

			Array begins:
=========================================================
m\n| 1  2    3     4      5       6        7         8
---|-----------------------------------------------------
1  | 1  2    3     4      5       6        7         8...
2  | 2  6   11    18     27      38       51        66...
3  | 3 11   48   109    218     405      724      1277...
4  | 4 18  109   632   1649    4192    10889     29480...
5  | 5 27  218  1649  10130   34801   116772    402673...
6  | 6 38  405  4192  34801  194292   856225   3804880...
7  | 7 51  724 10889 116772  856225  4730810  24810465...
8  | 8 66 1277 29480 402673 3804880 24810465 145114944...
...

Links

Crossrefs

Main diagonal is A291104.
Cf. A008299, A290586, A290632, A290818.

Programs

  • Mathematica
    T32[n_, k_] := n^k + k^n - Min[n, k]!*StirlingS2[Max[n, k], Min[n, k]];
T99[n_, k_] := Sum[(-1)^i*Binomial[n, i]*Sum[(-1)^j*((k - i - j)^(n - i)/(j!*(k - i - j)!)), {j, 0, k - i}], {i, 0, k}];
T[m_, n_] /; n >= m := T32[m, n] + Sum[Sum[Binomial[m, k]*Binomial[n, j]*k!*T99[n - j, k - 1]*j!*StirlingS2[m - k, j - 1], {j, 2, m - k}], {k, 2, m - 2}]; T[m_, n_] /; n < m := T[n, m];
Table[T[m - n + 1, n], {m, 1, 10}, {n, 1, m}] // Flatten(* Jean-François Alcover, Nov 01 2017, after Andrew Howroyd *)
  • PARI
    \\ here s(n,k) is A008299(n,k) and b(m,n,1) is A290632(m,n).
s(n, k)=sum(i=0, min(n, k), (-1)^i * binomial(n, i) * stirling(n-i, k-i, 2) );
b(m, n, x) = m^n*x^n + n^m*x^m - if(n<=m, n!*x^m*stirling(m, n, 2), m!*x^n*stirling(n, m, 2));
p(m, n, x)={sum(k=2, m-2, sum(j=2, m-k, binomial(m, k) * binomial(n, j) * k! * s(n-j, k-1) * j! * stirling(m-k, j-1, 2) * x^(m+n-j-k) ))}
T(m, n) = b(m, n, 1) + p(m, n, 1);

Formula

T(m,n) = A290632(m, n) + Sum_{k=2..m-2} Sum_{j=2..m-k} binomial(m,k) * binomial(n,j) * k! * A008299(n-j,k-1) * j! * stirling2(m-k,j-1).
Showing 1-4 of 4 results.

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