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 1-7 of 7 results.

A008285 Erroneous version of A342587.

Original entry on oeis.org

1, 1, 2, 1, 12, 6, 1, 86, 108, 24, 1, 840, 2310, 960, 120, 1, 11642, 65700, 42960, 9000, 720, 1, 227892, 2583126, 2510760, 712320, 90720, 5040, 1, 6285806, 142259628, 199424904, 71243760, 11481120, 987840, 40320
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Examples

			Triangle T(n,k) (with n >= 1 and 1 <= k <= n) begins as follows:
  1;
  1,      2;
  1,     12,       6;
  1,     86,     108,      24;
  1,    840,    2310,     960,    120;
  1,  11642,   65700,   42960,   9000,   720;
  1, 227892, 2583126, 2510760, 712320, 90720, 5040;
  ...

Links

Crossrefs

Cf. A000142 (diagonal), A001035 (row sums), A055531 (k=2), A055532 (k=3), A055533 (subdiagonal), A081064, A342501 (connected).

A055533 Number of labeled order relations on n nodes in which longest chain has n-1 nodes.

Original entry on oeis.org

1, 12, 108, 960, 9000, 90720, 987840, 11612160, 146966400, 1995840000, 28979596800, 448345497600, 7366565606400, 128152088064000, 2353813862400000, 45527990796288000, 925143000477696000
Offset: 2

Views

Author

N. J. A. Sloane, Jul 10 2000

Keywords

Comments

Also number of labeled acyclic digraphs with n nodes and binomial(n,2)-1 arcs, cf. A081064. - Vladeta Jovovic, Jul 14 2008

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

Crossrefs

A column or diagonal of triangle in A342587.

Programs

  • PARI
    a(n)=if(n<2,0,n!/2*(n-1)^2) /* Michael Somos, Mar 06 2004 */

Formula

E.g.f.: (x^2/2)(1+x)/(1-x)^3. a(n)=n!(n-1)^2/2, if n>1. - Vladeta Jovovic, Dec 01 2002
D-finite with recurrence (n-2)^2*a(n) -n*(n-1)^2*a(n-1)=0. - R. J. Mathar, Mar 08 2021

Extensions

More terms from Vladeta Jovovic, Dec 01 2002

A361951 Triangle read by rows: T(n,k) is the number of labeled weakly graded (ranked) posets with n elements and rank k.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 12, 6, 0, 1, 86, 108, 24, 0, 1, 840, 2190, 840, 120, 0, 1, 11642, 55620, 31800, 6840, 720, 0, 1, 227892, 1858206, 1428000, 384720, 60480, 5040, 0, 1, 6285806, 82938828, 80529624, 24509520, 4626720, 584640, 40320
Offset: 0

Views

Author

Andrew Howroyd, Mar 31 2023

Keywords

Comments

Here weakly graded means that there exists a rank function rk from the poset to the integers such that whenever v covers w in the poset, we have rk(v) = rk(w) + 1.
T(n,k) corresponds to a(k,n) = b(k,n) - b(k-1,n) in the Klarner reference. Figure 2 shows the posets of row n=4.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1,      2;
  0, 1,     12,       6;
  0, 1,     86,     108,      24;
  0, 1,    840,    2190,     840,    120;
  0, 1,  11642,   55620,   31800,   6840,   720;
  0, 1, 227892, 1858206, 1428000, 384720, 60480, 5040;
  ...

Links

Crossrefs

Row sums are A001833.
Column k=2 is A055531.
Partial row sums include A000007, A000012, A001831, A001832.
Main diagonal is A000142.
The unlabeled version is A361953.
Cf. A342587, A361950.

Programs

  • PARI
    \\ Here C(n) gives columns of A361950 as vector of e.g.f.'s.
S(M)={matrix(#M, #M, i, j, sum(k=0,  i-j, 2^((j-1)*k)*M[i-j+1,k+1])/(j-1)! )}
C(n,m=n)={my(M=matrix(n+1, n+1), c=vector(m+1), A=O(x*x^n)); M[1, 1]=1; c[1]=1+A; for(h=1, m, M=S(M); c[h+1]=sum(i=0, n, vecsum(M[i+1, ])*x^i, A)); c}
T(n)={my(c=C(n), b=vector(n+1, h, c[h]/c[max(h-1,1)])); Mat(vector(n+1, h, Col(serlaplace(b[h]-if(h>1, b[h-1])), -n-1)))}
{ my(A=T(7)); for(n=1, #A, print(A[n, 1..n])) }

Formula

E.g.f. of column k >=2: C(k,x)/C(k-1,x) - C(k-1,x)/C(k-2,x) where C(k,x) is the e.g.f. of column k of A361950.

A342501 T(n,k) is the number of connected labeled posets with n elements and rank k: triangle read by rows.

Original entry on oeis.org

1, 0, 2, 0, 6, 6, 0, 38, 84, 24, 0, 390, 1710, 840, 120, 0, 6062, 49740, 36840, 8280, 720, 0, 134526, 2050566, 2184000, 646800, 85680, 5040, 0, 4172198, 118645044, 177549624, 65313360, 10735200, 947520, 40320
Offset: 1

Views

Author

R. J. Mathar, Mar 14 2021

Keywords

Comments

This is a variant of A342587 admitting only connected posets.

Examples

			The table starts in row n=1 and shows ranks k>=0:
1: 1
2: 0 2
3: 0 6 6
4: 0 38 84 24
5: 0 390 1710 840 120
6: 0 6062 49740 36840 8280 720
7: 0 134526 2050566 2184000 646800 85680 5040
8: 0 4172198 118645044 177549624 65313360 10735200 947520 40320

Links

Crossrefs

Cf. A001927 (row sums), A000142 (diagonal), A002031/A002027 (rank 1), A342500 (unlabeled).

A055531 Number of labeled order relations on n nodes in which longest chain has 2 nodes.

Original entry on oeis.org

2, 12, 86, 840, 11642, 227892, 6285806, 243593040, 13262556722, 1014466283292, 109128015915206, 16521353903210520, 3524056001906654762, 1059868947134489801412, 449831067019305308555486, 269568708630308018001547680, 228228540531327778410439620962
Offset: 2

Views

Author

N. J. A. Sloane, Jul 10 2000

Keywords

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

Crossrefs

A column or diagonal of triangle in A342587.
Cf. A001831, A052296.

Formula

a(n) = A001831(n)-1.
a(n) = Sum_{k=1..n-1} binomial(n,k)*(2^(n-k)-1)^k. - Geoffrey Critzer, Oct 29 2023

Extensions

More terms from Vladeta Jovovic, Oct 24 2000
a(13)-a(16) corrected and more terms from Sean A. Irvine, Mar 25 2022

A055532 Number of labeled order relations on n nodes in which longest chain has 3 nodes.

Original entry on oeis.org

6, 108, 2310, 65700, 2583126, 142259628, 11012710470, 1196543891700, 181782466114326, 38435786111785788, 11256358984173551430
Offset: 3

Views

Author

N. J. A. Sloane, Jul 10 2000

Keywords

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

Crossrefs

A column or diagonal of triangle in A342587.

Extensions

Extended to n=13 by Brendan McKay, Mar 16 2021

A055534 Number of labeled order relations on n nodes in which longest chain has n-2 nodes.

Original entry on oeis.org

1, 86, 2310, 42960, 712320, 11481120, 186671520, 3116534400, 53907638400, 970417324800, 18217668268800
Offset: 3

Views

Author

N. J. A. Sloane, Jul 10 2000

Keywords

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

Crossrefs

A column or diagonal of triangle in A342587.

Extensions

Extended to n=13 by Brendan McKay, Mar 16 2021
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.