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 1-6 of 6 results.

A001827 Related to graded partially ordered sets.

Original entry on oeis.org

1, 4, 22, 166, 1726, 24814, 494902, 13729846, 531077086, 28697950174, 2170176736102, 230007989092006, 34211282155446286, 7149766552058591374, 2101690590380890192342, 869808621195903097079446, 507261036269544624540347326
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Corresponds to the numbers c(4,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=4 of A361950.
Cf. A000684, A001828, A001829, A001830, A047863.

Formula

a(n) = Sum_{p+q+r+s=n} (n!/p!q!r!s!) 2^(pq+qr+rs) where (p,q,r,s) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

Extensions

More terms from Sean A. Irvine, Sep 24 2015

A001828 Related to graded partially ordered sets.

Original entry on oeis.org

1, 5, 33, 287, 3309, 50975, 1058493, 29885567, 1156711869, 61815727295, 4589058616413, 475576073939807, 69061902766811229, 14093318360697120095, 4049931601653596366013, 1641314561238334948886207, 939097032426474389539281789
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Corresponds to the numbers c(5,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=5 of A361950.
Cf. A000684, A001827, A001829, A001830, A047863.

Formula

a(n) = Sum_{p+q+r+s+t=n} (n!/p!q!r!s!t!) 2^(pq+qr+rs+st) where (p,q,r,s,t) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

Extensions

More terms from Sean A. Irvine, Sep 24 2015

A001829 Related to graded partially ordered sets.

Original entry on oeis.org

1, 6, 46, 450, 5650, 91866, 1957066, 55363650, 2109599650, 109773407466, 7894945079386, 792252362302770, 111671194813402930, 22202849561274787866, 6241728810901739517226, 2484011055161613143144610, 1400187830319472451472442690
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Corresponds to the numbers c(6,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=6 of A361950.
Cf. A000684, A001827, A001828, A001830, A047863.

Formula

a(n) = Sum_{p+q+r+s+t+u=n} (n!/p!q!r!s!t!u!) 2^(pq+qr+rs+st+tu) where (p,q,r,s,t,u) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

A001830 Related to graded partially ordered sets.

Original entry on oeis.org

1, 7, 61, 661, 8953, 152917, 3334921, 94354981, 3528929353, 177999003157, 12340001650921, 1194005625114661, 162936187792764073, 31536761103831315157, 8677703806537883683081, 3395880602480076153665701, 1889190751946097573211698313
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Corresponds to the numbers c(7,n) in the Klarner paper. - Sean A. Irvine, Sep 24 2015

References

  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column k=7 of A361950.
Cf. A000684, A001827, A001828, A001829, A047863.

Formula

a(n) = Sum_{p+q+r+s+t+u+v=n} (n!/p!q!r!s!t!u!v!) 2^(pq+qr+rs+st+tu+uv) where (p,q,r,s,t,u,v) is any nonnegative composition of n. - Sean A. Irvine, Sep 24 2015

Extensions

More terms from Sean A. Irvine, Sep 24 2015

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.

A361952 Array read by antidiagonals: T(n,k) is the number of unlabeled posets with n elements together with a function rk mapping each element to a rank between 1 and k such that whenever v covers w in the poset then rk(v) = rk(w) + 1.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 8, 1, 0, 1, 5, 13, 21, 17, 1, 0, 1, 6, 19, 40, 58, 38, 1, 0, 1, 7, 26, 66, 126, 172, 94, 1, 0, 1, 8, 34, 100, 228, 420, 569, 258, 1, 0, 1, 9, 43, 143, 373, 816, 1537, 2148, 815, 1, 0, 1, 10, 53, 196, 571, 1412, 3140, 6342, 9538, 3038, 1, 0
Offset: 0

Views

Author

Andrew Howroyd, Mar 31 2023

Keywords

Comments

A poset is counted once for each admissible ranking function. This is an intermediate step in the computation of A361953 where each graded poset is counted exactly once.

Examples

			Array begins:
============================================
n/k| 0 1   2    3    4     5     6     7 ...
---+----------------------------------------
0  | 1 1   1    1    1     1     1     1 ...
1  | 0 1   2    3    4     5     6     7 ...
2  | 0 1   4    8   13    19    26    34 ...
3  | 0 1   8   21   40    66   100   143 ...
4  | 0 1  17   58  126   228   373   571 ...
5  | 0 1  38  172  420   816  1412  2272 ...
6  | 0 1  94  569 1537  3140  5631  9351 ...
7  | 0 1 258 2148 6342 13383 24410 41097 ...
  ...

Links

Crossrefs

Columns k=0..2 are A000007, A000012, A049312.
Rows n=0..4 are A000012, A000027, A034856, A137742.
The labeled version is A361950.
Cf. A361953.

Programs

  • PARI
    \\ See Links in A361953 for program.
{ my(A=A361952tabl(7)); for(i=1, #A, print(A[i,])) }
Showing 1-6 of 6 results.

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

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