A253186 -id:A253186 - cp's OEIS frontend

A034347 Number of binary [ n,6 ] codes without 0 columns.

0, 0, 0, 0, 0, 1, 6, 25, 99, 385, 1472, 5676, 22101, 87404, 350097, 1413251, 5708158, 22903161, 90699398, 352749035, 1342638839, 4990325414, 18090636016, 63933709870, 220277491298, 740170023052, 2426954735273, 7770739437179, 24314436451415, 74406425640743, 222867051758565, 653898059035166

Offset: 1

N. J. A. Sloane

nonn

Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k=6.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = S_{n,6,2}.]
Petros Hadjicostas, Generating function for a(n).
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.

Cf. A034254, A034344, A034345, A034346, A034348, A034349, A253186.
First differences of A034360.
Column k = 6 of A034253.

# Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
def A034253col(k, length):
    G1 = PSL(k, GF(2))
    G2 = PSL(k-1, GF(2))
    D1 = G1.cycle_index()
    D2 = G2.cycle_index()
    f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
    f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
    f = f1 - f2
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 6 (this sequence) gives
print(A034253col(6, 30)) # Petros Hadjicostas, Oct 05 2019

More terms from Petros Hadjicostas, Oct 05 2019

A034348 Number of binary [ n,7 ] codes without 0 columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 7, 35, 170, 847, 4408, 24297, 143270, 901491, 5985278, 41175203, 287813284, 2009864185, 13848061942, 93369988436, 613030637339, 3908996099141, 24179747870890, 145056691643428, 844229016035010, 4769751989333029, 26181645303024760, 139750488576152520

Offset: 1

N. J. A. Sloane

nonn

To find the g.f., modify the Sage program below (cf. function f). It is very complicated to write it here. - Petros Hadjicostas, Oct 05 2019

Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k=7.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = S_{n,7,2}.]
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.

Cf. A034254, A034344, A034345, A034346, A034347, A034349, A253186.

Column k=7 of A034253 and first differences of A034361.

# Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
def A034253col(k, length):
    G1 = PSL(k, GF(2))
    G2 = PSL(k-1, GF(2))
    D1 = G1.cycle_index()
    D2 = G2.cycle_index()
    f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
    f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
    f = f1 - f2
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 7 (this sequence) gives
print(A034253col(7, 30)) #

More terms from Petros Hadjicostas, Oct 05 2019

A034349 Number of binary [ n,8 ] codes without 0 columns.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 8, 47, 277, 1775, 12616, 102445, 957357, 10174566, 119235347, 1482297912, 18884450721, 240477821389, 3012879828566, 36800049400028, 436068618826236, 5001537857507095, 55482177298724426, 595303034603214108, 6181562837200509792, 62170512250565592346

Offset: 1

N. J. A. Sloane

nonn

To find the g.f., modify the Sage program below (cf. function f). It is very complicated to write it here. - Petros Hadjicostas, Oct 07 2019

Discrete algorithms at the University of Bayreuth, Symmetrica.
Harald Fripertinger, Isometry Classes of Codes.
Harald Fripertinger, Snk2: Number of the isometry classes of all binary (n,k)-codes without zero-columns. [See column k=8.]
H. Fripertinger and A. Kerber, Isometry classes of indecomposable linear codes. In: G. Cohen, M. Giusti, T. Mora (eds), Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 11th International Symposium, AAECC 1995, Lect. Notes Comp. Sci. 948 (1995), pp. 194-204. [Here a(n) = S_{n,8,2}.]
Petr Lisonek, Combinatorial families enumerated by quasi-polynomials, J. Combin. Theory Ser. A 114(4) (2007), 619-630. [See Section 5.]
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
David Slepian, Some further theory of group codes, Bell System Tech. J. 39(5) (1960), 1219-1252.
Wikipedia, Cycle index.
Wikipedia, Projective linear group.

Cf. A034254, A034344, A034345, A034346, A034347, A034348, A253186.

Column k=8 of A034253 and first differences of A034362.

# Fripertinger's method to find the g.f. of column k >= 2 of A034253 (for small k):
def A034253col(k, length):
    G1 = PSL(k, GF(2))
    G2 = PSL(k-1, GF(2))
    D1 = G1.cycle_index()
    D2 = G2.cycle_index()
    f1 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D1)
    f2 = sum(i[1]*prod(1/(1-x^j) for j in i[0]) for i in D2)
    f = f1 - f2
    return f.taylor(x, 0, length).list()
# For instance the Taylor expansion for column k = 8 (current sequence) gives
print(A034253col(8, 30)) # Petros Hadjicostas, Oct 07 2019

More terms from Petros Hadjicostas, Oct 07 2019

A342059 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of 2-connected planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 3, k=2..2*n-4.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 5, 2, 1, 1, 3, 17, 31, 22, 6, 2, 1, 4, 42, 157, 318, 265, 123, 26, 6, 1, 6, 87, 576, 2128, 4009, 4055, 2332, 804, 147, 17, 1, 7, 161, 1664, 9659, 31252, 59244, 66289, 46521, 20604, 5743, 892, 73, 1, 9, 286, 4151, 34700, 168757, 505410, 952044, 1156127, 931227, 506318, 183980, 43180, 5876, 389

Offset: 3

Andrew Howroyd, Mar 27 2021

The number of edges is n+k-2.

Terms of this sequence can be computed using the tool "plantri". The expanded reference gives rows 3..15 of this table.

			Triangle begins:
  1;
  1, 1,  1;
  1, 2,  5,   2,    1;
  1, 3, 17,  31,   22,    6,    2;
  1, 4, 42, 157,  318,  265,  123,   26,   6;
  1, 6, 87, 576, 2128, 4009, 4055, 2332, 804, 147, 17;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..171 (rows 3..15)
Gunnar Brinkmann and Brendan McKay, Fast generation of planar graphs (expanded edition), Tables 23-26.

Row sums are A342058.

Cf. A006406 (by edges), A239893 (3-connected), A342060.

T(n,2) = 1.

T(n,3) = A253186(n-2).

A309936 Irregular triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs with n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 1, 2, 3, 1, 1, 0, 1, 3, 7, 6, 4, 1, 1, 0, 1, 4, 13, 17, 17, 8, 4, 1, 1, 0, 1, 6, 25, 44, 56, 41, 24, 9, 4, 1, 1, 0, 1, 7, 40, 101, 164, 158, 117, 57, 26, 9, 4, 1, 1, 0, 1, 9, 65, 216, 450, 562, 503, 315, 162, 64, 27, 9, 4, 1, 1

Offset: 1

Andrew Howroyd, Oct 23 2019

Covering k vertices means there are no vertices of degree zero.

			Triangle begins:
  0, 1;
  0, 1, 1,  1;
  0, 1, 2,  3,   1,   1;
  0, 1, 3,  7,   6,   4,   1,   1;
  0, 1, 4, 13,  17,  17,   8,   4,  1,  1;
  0, 1, 6, 25,  44,  56,  41,  24,  9,  4, 1, 1;
  0, 1, 7, 40, 101, 164, 158, 117, 57, 26, 9, 4, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..650 (rows 1..25)

Row sums are A050535.
Columns k=3..4 are A253186, A328652.
Cf. A191646, A192517, A327615.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}
C(n,m)={my(s=O(x*x^m)); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i+O(x*x^m))); Col(s/n!)}
T(m) = {my(n=2*m, A=Mat(vector(n+1, n, C(n-1,m)))); A[2..m+1, 2..n+1]-A[2..m+1, 1..n]}
{ my(A=T(8)); for(n=1, matsize(A)[1], print(A[n, 1..2*n])) }

T(n,k) = A192517(k,n) - A192517(k-1,n) for k > 1.

A343870 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) planar graphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 3, 3, 1, 0, 0, 0, 0, 2, 9, 4, 1, 0, 0, 0, 0, 1, 13, 20, 6, 1, 0, 0, 0, 0, 0, 11, 49, 40, 7, 1, 0, 0, 0, 0, 0, 5, 77, 158, 70, 9, 1, 0, 0, 0, 0, 0, 2, 75, 406, 426, 121, 11, 1, 0, 0, 0, 0, 0, 0, 47, 662, 1645, 1018, 189, 13, 1, 0

Offset: 1

Andrew Howroyd, May 04 2021

			Triangle T(n,k) begins (n edges >= 1, k vertices >= 2):
  1;
  0, 0;
  0, 1, 0;
  0, 0, 1, 0;
  0, 0, 1, 1,  0;
  0, 0, 1, 2,  1,  0;
  0, 0, 0, 3,  3,  1,   0;
  0, 0, 0, 2,  9,  4,   1,   0;
  0, 0, 0, 1, 13, 20,   6,   1,   0;
  0, 0, 0, 0, 11, 49,  40,   7,   1,  0;
  0, 0, 0, 0,  5, 77, 158,  70,   9,  1, 0;
  0, 0, 0, 0,  2, 75, 406, 426, 121, 11, 1, 0;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..351 (26 rows) (first 210 terms (20 rows) from Andrew Howroyd)

Row sums are A343869.
Column sums are A021103.
Cf. A049334, A049336 (transpose), A049337, A253186, A339070.

geng -C $k $n:$n | planarg -q | countg -q # Georg Grasegger, Jun 05 2023

T(n, n) = 1 for n >= 3.

T(n, n-1) = A253186(n-3) for n >= 3.

A160138 a(n) = number of solutions to the system: x + y + z + w = n, -2x - y + z + 2w = 5 with nonnegative x, y, z, w.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 6, 7, 9, 11, 13, 15, 18, 20, 23, 26, 29, 32, 36, 39, 43, 47, 51, 55, 60, 64, 69, 74, 79, 84, 90, 95, 101, 107, 113, 119, 126, 132, 139, 146, 153, 160, 168, 175, 183, 191, 199, 207, 216, 224, 233, 242, 251, 260, 270, 279, 289, 299, 309, 319, 330, 340

Offset: 1

Krishnan Sundararaman (krishnan.sundararaman(AT)enmu.edu), May 02 2009

nonn

Number of ways in which playing n one-card-poker games results in a payoff of $5.
x = # of games where player loses $2,
y = # of games where player loses $1,
z = # of games where player wins $1,
w = # of games where player wins $2.
The events i.e. winning $1, losing $2 etc. are mutually exclusive.
Hence in n games
x+y+z+w = n
-2x-y+z+2w = $5

			For n = 3, a(3) = 1, since the four-tuple <x=0, y=0, z=1, w=2> allows you to win $5 in 3 games. x + y + z + w = 1 + 2 =3, -2x - y + z + 2w = 1 + 2*2 = 5, as desired.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
One Card Poker
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).

Cf. A008806 is the number of ways in which playing n one-card-poker games results in a payoff of $0, i.e., the n-th term is the number of solutions to the system: x + y + z + w = n, -2x - y + z + 2w = 0, with nonnegative x, y, z, w.

Except for offset, same as A253186.

> fourples2 := proc (n) local i, c1, c2, c3, c4, c3positive, mylist, cash, k, howmanyways; cash := 2*n; for k from -cash to cash do i := 0; unassign(mylist); for c1 from 0 to n do c3positive := true; for c2 from 0 to n-c1 while c3positive do c3 := 2*n-4*c1-3*c2-k; if 0 <= c3 then c4 := n-c1-c2-c3; if 0 <= c4 then i := i+1; mylist[i] := [c1, c2, c3, c4] end if else c3positive := false end if end do end do; howmanyways[k] := [i, [seq(mylist[j], j = 1 .. i)]] end do; return howmanyways end proc; N := 20; for n to N do a := fourples2(n); points[n] := [n, a[5][1]] end do; seq(points[n], n = 1 .. N);

LinearRecurrence[{1, 1, 0, -1, -1, 1}, {0, 0, 1, 2, 3, 4}, 100] (* Jean-François Alcover, Apr 11 2020 *)

concat([0, 0], Vec((1 + x - x^3)/((1 - x)^3*(1 + x)*(1 + x + x^2)) + O(x^60))) \\ Andrew Howroyd, Jan 12 2020

From Andrew Howroyd, Jan 12 2020: (Start)
a(n) = A253186(n-1).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6) for n > 6.
G.f.: x^3*(1 + x - x^3)/((1 - x)^3*(1 + x)*(1 + x + x^2)).
(End)

Terms a(13) and beyond from Andrew Howroyd, Jan 12 2020

cp's OEIS Frontend

A034347 Number of binary [ n,6 ] codes without 0 columns.

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SageMath

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A034348 Number of binary [ n,7 ] codes without 0 columns.

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Sage

Extensions

A034349 Number of binary [ n,8 ] codes without 0 columns.

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Sage

Extensions

A342059 Triangle read by rows: T(n,k) is the number of embeddings on the sphere of 2-connected planar graphs with n nodes and k faces up to orientation preserving isomorphisms, n >= 3, k=2..2*n-4.

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A309936 Irregular triangle read by rows: T(n,k) is the number of unlabeled loopless multigraphs with n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

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A343870 Triangle read by rows: T(n,k) is the number of unlabeled nonseparable (or 2-connected) planar graphs with n edges and k nodes (n >= 1, 2 <= k <= n + 1).

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nauty

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A160138 a(n) = number of solutions to the system: x + y + z + w = n, -2x - y + z + 2w = 5 with nonnegative x, y, z, w.

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