A055192 -id:A055192 - cp's OEIS frontend

A055182 n-th distinct number to appear in A055174.; also the n-th to appear in A055192.

4, 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, 14, 13, 17, 11, 15, 19, 16, 22, 25, 27, 20, 24, 29, 21, 18, 26, 32, 23, 36, 28, 40, 31, 30, 43, 34, 46, 35, 49, 37, 52, 42, 38, 33, 55, 45, 44, 41, 39, 58, 50, 63, 53, 66, 57, 54, 48, 69, 59, 74, 62, 60, 51

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Conjecture: this sequence is a permutation of the positive integers.

A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

Original entry on oeis.org

1, 3, 6, 16, 34, 90, 211, 558, 1430, 3908, 10725, 30825, 90156, 273234, 848355, 2714399, 8909057, 30042866, 103859678, 368075596, 1335537312, 4958599228, 18820993913, 72980867400, 288885080660, 1166541823566, 4802259167367, 20141650236664

Offset: 1

Peter J. Cameron

Also the number of bipartite graphs with n edges, no isolated vertices and a distinguished bipartite block, up to isomorphism.
The EULERi transform (A056156) is also interesting.
a(n) is also the number of non-isomorphic set multipartitions (multisets of sets) of weight n. - Gus Wiseman, Mar 17 2017

			E.g. a(2) = 3: two ones in same row, two ones in same column, or neither.
a(3) = 6 is coefficient of x^3 in (1/36)*((1 + x)^9 + 6*(1 + x)^3*(1 + x^2)^3 + 8*(1 + x^3)^3 + 9*(1 + x)*(1 + x^2)^4 + 12*(1 + x^3)*(1 + x^6))=1 + x + 3*x^2 + 6*x^3 + 7*x^4 + 7*x^5 + 6*x^6 + 3*x^7 + x^8 + x^9.
There are a(3) = 6 binary matrices with 3 ones, with no zero rows or columns, up to row and column permutation:
  [1 0 0] [1 1 0] [1 0] [1 1] [1 1 1] [1]
  [0 1 0] [0 0 1] [1 0] [1 0] ....... [1].
  [0 0 1] ....... [0 1] ............. [1]
Non-isomorphic representatives of the a(3)=6 set multipartitions are: ((123)), ((1)(23)), ((2)(12)), ((1)(1)(1)), ((1)(2)(2)), ((1)(2)(3)). - _Gus Wiseman_, Mar 17 2017

Aliaksandr Siarhei, Table of n, a(n) for n = 1..102
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Peter J. Cameron, D. A. Gewurz and F. Merola, Product action, Discrete Math., 308 (2008), 386-394.
Peter J. Cameron, Problems on Permutation Groups, see Problem 3
Index entries for sequences related to binary matrices.

Main diagonal of A321609.
Cf. A049312, A048194, A028657, A055192, A055599, A052371, A052370, A053304, A053305, A007716, A002724.
Cf. A057149, A057150, A057151, A057152.
Cf. A034691, A056156, A089259, A116540, A283877.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
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}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 16 2023

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, then apply EULER transform.

a(n) is the coefficient of x^n in the cycle index Z(S_n X S_n; 1+x, 1+x^2, ...), where S_n X S_n is Cartesian product of symmetric groups S_n of degree n.

More terms and formula from Vladeta Jovovic, Jul 29 2000
a(19)-a(28) from Max Alekseyev, Jul 22 2009
a(29)-a(102) from Aliaksandr Siarhei, Dec 13 2013
Name edited by Gus Wiseman, Dec 18 2018

A007776 Number of connected posets with n elements of height 1.

Original entry on oeis.org

1, 2, 4, 10, 27, 88, 328, 1460, 7799, 51196, 422521, 4483460, 62330116, 1150504224, 28434624153, 945480850638, 42417674401330, 2572198227615998, 211135833162079184, 23487811567341121158, 3545543330739039981738, 727053904070651775719646

Offset: 2

Georg Wambach (gw(AT)informatik.Uni-Koeln.de)

nonn

Inverse Euler transform of A048194 and A049312. - Detlef Pauly (dettodet(AT)yahoo.de) and Vladeta Jovovic, Jul 25 2003
Essentially the same as A318870. - Georg Fischer, Oct 02 2018
Number of connected digraphs on n unlabeled nodes where every node has indegree 0 or outdegree 0 and there are no isolated nodes.  - Andrew Howroyd, Oct 03 2018

Andrew Howroyd, Table of n, a(n) for n = 2..50 (terms 2..40 from Alois P. Heinz)
N. J. A. Sloane, Transforms
J. Textor, A. Idelberger, and M. Liskiewicz, Learning from Pairwise Marginal Independencies, arXiv:1508.00280 [cs.AI], 2015.
Index entries for sequences related to posets

Cf. A005142, A002031 (labeled case), A048194, A049312, A055192, A318870, column 1 of A342500.

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
b[d_] := Sum[A[n, d - n], {n, 0, d}];
EULERi[Array[b, 30]] // Rest (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A049312 *)

Inverse Euler transform of A055192. - Andrew Howroyd, Oct 03 2018

More terms from Vladeta Jovovic, Jul 25 2003

Offset corrected by Andrew Howroyd, Oct 03 2018

A056152 Triangular array giving number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k=1..n-1 vertices, up to isomorphism.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 8, 17, 8, 1, 1, 11, 42, 42, 11, 1, 1, 15, 91, 179, 91, 15, 1, 1, 19, 180, 633, 633, 180, 19, 1, 1, 24, 328, 2001, 3835, 2001, 328, 24, 1, 1, 29, 565, 5745, 20755, 20755, 5745, 565, 29, 1, 1, 35, 930, 15274, 102089, 200082, 102089

Offset: 2

Vladeta Jovovic, Jul 29 2000

Also table read by rows: for 0 < k < n, a(n, k) = number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k vertices, up to isomorphism.
a(n, k) is the number of isomorphism classes of finite subdirectly irreducible almost distributive lattices in which the N-quotient has k upper covers and (n - k) lower covers. - David Wasserman, Feb 11 2002
Also, row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color, with no isolated nodes; the color classes are not interchangeable.

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   5,   1;
  1,  8,  17,   8,  1;
  1, 11,  42,  42,  11,  1;
  1, 15,  91, 179,  91,  15,  1;
  1, 19, 180, 633, 633, 180, 19, 1;
  ...
There are 17 bipartite graphs with 6 vertices, no isolated vertices and a distinguished bipartite block with 3 vertices, or equivalently, there are 17 3 X 3 binary matrices with no zero rows or columns, up to row and column permutation:
[0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]
[0 0 1] [0 0 1] [0 1 0] [0 1 0] [0 1 0] [0 1 1] [0 1 1] [0 1 1] [1 1 0]
[1 1 0] [1 1 1] [1 0 0] [1 0 1] [1 1 1] [1 0 1] [1 1 0] [1 1 1] [1 1 0]
and
[0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
[1 1 0] [1 1 1] [0 1 1] [0 1 1] [1 0 1] [1 0 1] [1 1 1] [1 1 1]
[1 1 1] [1 1 1] [1 0 1] [1 1 1] [1 1 0] [1 1 1] [1 1 1] [1 1 1].

J. G. Lee, Almost Distributive Lattice Varieties, Algebra Universalis, 21 (1985), 280-304.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43. See Table 2.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy] See Table 2.

Columns k=1..6 are A000012, A024206, A055609, A055082, A055083, A055084.
Row sums give A055192.
See A122083 for another version of this triangle.
Cf. A049312, A048194, A028657, A049311.

A361957 Triangle read by rows: T(n,k) is the number of unlabeled tiered posets with n elements and height k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 12, 10, 4, 1, 0, 1, 35, 35, 16, 5, 1, 0, 1, 108, 149, 66, 23, 6, 1, 0, 1, 393, 755, 327, 106, 31, 7, 1, 0, 1, 1666, 4736, 1936, 566, 156, 40, 8, 1, 0, 1, 8543, 37394, 14130, 3578, 878, 217, 50, 9, 1

Offset: 0

Andrew Howroyd, Apr 03 2023

A tiered poset is a partially ordered set in which every maximal chain has the same length.

			Triangle begins:
  1;
  0, 1;
  0, 1,    1;
  0, 1,    2,    1;
  0, 1,    5,    3,    1;
  0, 1,   12,   10,    4,   1;
  0, 1,   35,   35,   16,   5,   1;
  0, 1,  108,  149,   66,  23,   6,  1;
  0, 1,  393,  755,  327, 106,  31,  7, 1;
  0, 1, 1666, 4736, 1936, 566, 156, 40, 8, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..860 (rows 0..40).
Andrew Howroyd, PARI Program, Apr 2023.

Row sums are A361912.
Column k=2 is A055192.
The labeled version is A361956.
Cf. A361953, A361958 (connected).

\\ See link for program code.
{ my(A=A361957tabl(9)); for(i=1, #A, print(A[i, 1..i])) }

A122083 Triangle read by rows in which row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color, with no isolated nodes; the color classes are not interchangeable.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 3, 1, 0, 0, 1, 5, 5, 1, 0, 0, 1, 8, 17, 8, 1, 0, 0, 1, 11, 42, 42, 11, 1, 0, 0, 1, 15, 91, 179, 91, 15, 1, 0, 0, 1, 19, 180, 633, 633, 180, 19, 1, 0, 0, 1, 24, 328, 2001, 3835, 2001, 328, 24, 1, 0, 0, 1, 29, 565, 5745, 20755, 20755

Offset: 0

N. J. A. Sloane, Oct 19 2006

			K M N Gives the number N of unlabeled bicolored graphs with no isolated nodes and having K nodes of one color and M nodes of the other color.
0 0 1
Total( 0)= 1
0 1 0
1 0 0
Total( 1)= 0
0 2 0
1 1 1
2 0 0
Total( 2)= 1
0 3 0
1 2 1
2 1 1
3 0 0
Total( 3)= 2
0 4 0
1 3 1
2 2 3
3 1 1
4 0 0
Total( 4)= 5
0 5 0
1 4 1
2 3 5
3 2 5
4 1 1
5 0 0
Total( 5)= 12
0 6 0
1 5 1
2 4 8
3 3 17
4 2 8
5 1 1
6 0 0
Total( 6)= 35

J. G. Lee, Almost Distributive Lattice Varieties, Algebra Universalis, 21 (1985), 280-304.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

R. W. Robinson, First 20 rows, flattened
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43. See Table 2.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy] See Table 2.

Row sums give A055192. See A056152 for a version of this triangle with the bounding zeros in each row.

A055181 Number of new numbers in n-th segment of A055174; see example line of A055174.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 4, 3, 2, 2, 3, 2, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 3, 2, 4, 2, 7, 4

Offset: 1

Clark Kimberling, Apr 27 2000

nonn

Also, the number of new numbers in n-th segment of A055192.

cp's OEIS Frontend

A055182 n-th distinct number to appear in A055174.; also the n-th to appear in A055192.

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A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

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PARI

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A007776 Number of connected posets with n elements of height 1.

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Mathematica

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A056152 Triangular array giving number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k=1..n-1 vertices, up to isomorphism.

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A361957 Triangle read by rows: T(n,k) is the number of unlabeled tiered posets with n elements and height k.

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PARI

A122083 Triangle read by rows in which row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color, with no isolated nodes; the color classes are not interchangeable.

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Examples

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A055181 Number of new numbers in n-th segment of A055174; see example line of A055174.

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