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.

A223911 Number of tiered orders on n nodes (corrected version of A006860).

Original entry on oeis.org

1, 1, 3, 13, 111, 1381, 25623, 678133, 26169951, 1447456261, 114973232583, 13034314621813, 2103826463800911, 481932523110975301, 156356753093586913143, 71729530379673590609653, 46471511649877647638694591, 42487759521494442057018000901, 54781291469300608901184153800103
Offset: 0

Views

Author

Joerg Arndt, Mar 29 2013, using information provided by Joel B. Lewis, M. F. Hasler and Michel Marcus in A006860

Keywords

Links

Crossrefs

Row sums of A361956.
Cf. A218695, A361912 (unlabeled version).

Programs

  • Maple
    f:= (i, j)-> add((-1)^(i-k)*binomial(i, k) *(2^k-1)^j, k=0..i):
b:= proc(n, i) option remember;
      `if`(n=0, 1, add(b(n-j, j)/j!*f(i, j), j=1..n))
    end:
a:= n-> n!*b(n, 1):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 23 2013
  • Mathematica
    f[i_, j_] := Sum[(-1)^(i-k)*Binomial[i, k]*(2^k-1)^j, {k, 0, i}]; b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, j]/j!*f[i, j], {j, 1, n}]]; a[n_] := n!*b[n, 1]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 08 2015, after Alois P. Heinz *)
  • PARI
    f(m,n) = sum(k=0, m, (-1)^(m-k) * binomial(m,k) * (2^k-1)^n );
mn(n,V,m) = n! / prod(k=1, m, V[k]! ); /* multinomial of V[1..m] */
A223911(n)=
{
    my(m=n, C=vector(n,j,1), z, t, ret);
    while ( 1,  /* for all compositions C[1..m] of n */
        t = mn(n,C,m) * prod(j=1, m-1, f(C[j],C[j+1]) );
        ret += t;
        if ( m<=1, break() ); /* last composition? */
        /* create next composition: */
        C[m-1] += 1;
        z = C[m];
        C[m] = 1;
        m += z - 2;
    );
    return(ret);
}
  • PARI
    \\ here f(m,n) is A218695.
f(m,n) = {sum(k=0, m, (-1)^(m-k) * binomial(m, k) * (2^k-1)^n )}
seq(n)={my(N=matrix(n,n,i,j, f(i,j)), T=vector(n), v=vector(n+1)); v[1]=1; for(r=1, n, T[r]=vector(r, k, (r==k) + binomial(r,k)*sum(i=1, r-k, T[r-k][i]*N[i,k])); v[1+r]=vecsum(T[r])); v} \\ Andrew Howroyd, Mar 29 2023

Formula

a(n) = sum(all composition C of n, M(C) * prod(j=1..m-1, f(C[j]*C[j+1]) ) ) where m is the number of parts of the current composition P, f(i,j) = sum(k=0..i, (-1)^(i-k) * binomial(i,k) * (2^k-1)^j ), and M(C) is the multinomial coefficient n!/prod(j=1..m, C[j]! ); see Pari code.
Klarner incorrectly gives prod(j=1..m-1, f(C[j]*C[m]) ) in the formula for a(n).
Conjecture: a(n) ~ c * 2^(n^2/4 + 3*n/2) / sqrt(n), where c = EllipticTheta[3, 0, 1/2^(1/4)] / (sqrt(Pi) * 2^(1/4)) = 2.020039... (based on the numerical analysis of 600 terms). - Vaclav Kotesovec, Apr 10 2015

Extensions

Added a(0) = 1 by Alois P. Heinz, Jul 23 2013

A361920 Number of unlabeled ranked posets with n elements.

Original entry on oeis.org

1, 1, 2, 5, 16, 61, 280, 1501, 9394, 68647, 591570, 6108298, 77162708, 1219779207, 24648006828, 647865966973, 22437052221282, 1032905858402302, 63591727342096158, 5258562027225785955, 586001891321599337103, 88241281449605821921186, 17996565026907866304071630
Offset: 0

Views

Author

Martin Rubey, Mar 29 2023

Keywords

Comments

A partially ordered set is ranked if there is a function from the poset elements to the integers such that the function value of a covering element is precisely one larger than the function value of the covered element. This is called graded by some authors.

Examples

			For n=5, A000112(n) - a(n) = 63 - 61 = 2 because we have 2 posets with 5 elements that are not ranked: a<b<c<d  a<e<d  and  a<c<e  a<d  b<d  b<e where < means "is covered by". - _Geoffrey Critzer_, Oct 29 2023

Links

Crossrefs

Row sums of A361953.
Cf. A000112, A001833 (labeled), A223911, A361912, A361955.

Programs

  • PARI
    \\ See PARI link in A361953 for program code.
A361920seq(20) \\ Andrew Howroyd, Apr 01 2023
  • Sage
    sum(1 for P in posets(n) if P.is_ranked())

Extensions

Terms a(8) and beyond from Andrew Howroyd, Mar 31 2023

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

Views

Author

Andrew Howroyd, Apr 03 2023

Keywords

Comments

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

Examples

			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;
  ...

Links

Crossrefs

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

Programs

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

A361959 Number of connected unlabeled tiered posets with n elements.

Original entry on oeis.org

1, 1, 1, 3, 8, 25, 83, 330, 1538, 8834, 63723, 591313, 7180087, 115411088, 2472336977, 70840464484, 2720170935989, 140167035897433, 9702508891845883, 903305670556552694, 113233954374954246849, 19137449587197829494989, 4366006746295284220605007
Offset: 0

Views

Author

Andrew Howroyd, Apr 03 2023

Keywords

Comments

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

Links

Crossrefs

Row sums of A361958.
Cf. A361912 (not necessarily connected).

Programs

Showing 1-4 of 4 results.

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