A361953 -id:A361953 - cp's OEIS frontend

A361920 Number of unlabeled ranked posets with n elements.

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

Martin Rubey, Mar 29 2023

nonn

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.

			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

Andrew Howroyd, Table of n, a(n) for n = 0..40

Row sums of A361953.

Cf. A000112, A001833 (labeled), A223911, A361912, A361955.

\\ See PARI link in A361953 for program code.
A361920seq(20) \\ Andrew Howroyd, Apr 01 2023

sum(1 for P in posets(n) if P.is_ranked())

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

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

Andrew Howroyd, Mar 31 2023

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.

			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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50).
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
Wikipedia, Graded poset.

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.

\\ 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])) }

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.

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])) }

A361954 Triangle read by rows: T(n,k) is the number of unlabeled connected weakly graded (ranked) posets with n elements and rank k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 4, 5, 1, 0, 10, 23, 8, 1, 0, 27, 107, 56, 11, 1, 0, 88, 557, 388, 98, 14, 1, 0, 328, 3271, 2888, 839, 149, 17, 1, 0, 1460, 22424, 23900, 7512, 1470, 209, 20, 1, 0, 7799, 183273, 226807, 73405, 14715, 2308, 278, 23, 1

Offset: 1

Andrew Howroyd, Mar 31 2023

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.

			Triangle begins:
  1;
  0,   1;
  0,   2,    1;
  0,   4,    5,    1;
  0,  10,   23,    8,   1;
  0,  27,  107,   56,  11,   1;
  0,  88,  557,  388,  98,  14,  1;
  0, 328, 3271, 2888, 839, 149, 17, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..820 (rows 1..40)
Wikipedia, Graded poset.

Column k=2 is A007776.
Row sums are A361955.
Cf. A342500, A361953 (not necessarily connected).

\\ See PARI link in A361953 for program code.
{ my(A=A361954tabl(8)); for(i=1, #A, print(A[i, 1..i])) }

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

Andrew Howroyd, Mar 31 2023

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.

			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 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..860 (first 41 antidiagonals).

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

\\ See Links in A361953 for program.
{ my(A=A361952tabl(7)); for(i=1, #A, print(A[i,])) }

A361955 Number of unlabeled connected weakly graded (ranked) posets with n elements.

Original entry on oeis.org

1, 1, 1, 3, 10, 42, 202, 1146, 7493, 56996, 508609, 5414635, 70214227, 1134439731, 23331152887, 621768153861, 21761221300058, 1009759125475973, 62534859409597022, 5193886959561972984, 580677490292990902682, 87649885799470898359728, 17907726747155924589913398

Offset: 0

Andrew Howroyd, Mar 31 2023

nonn

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.

Andrew Howroyd, Table of n, a(n) for n = 0..40
Wikipedia, Graded poset.

Row sums of A361954.

Cf. A000608, A361920, A361953.

\\ See PARI link in A361953 for program code.
A361955seq(20)

Inverse Euler transform of A361920.

cp's OEIS Frontend

A361920 Number of unlabeled ranked posets with n elements.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Sage

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A361954 Triangle read by rows: T(n,k) is the number of unlabeled connected weakly graded (ranked) posets with n elements and rank k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A361955 Number of unlabeled connected weakly graded (ranked) posets with n elements.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula