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 1-3 of 3 results.

A375838 Number of rooted chains starting with the cycle (1)(2)(3)...(n) in the permutation poset of [n].

Original entry on oeis.org

1, 1, 2, 9, 83, 1270, 28799, 906899, 37866842, 2024422837, 134850653405, 10950546880152, 1064840930492393, 122158078221727119, 16325324374155336370, 2514183676808883419043, 442023695390488997377405, 87989953715757624724243004, 19688099473681895327628896249, 4919839221134662388853128069571, 1365091729320293490230304687026514
Offset: 0

Views

Author

Rajesh Kumar Mohapatra, Subhashree Sahoo, and Ranjan Kumar Dhani, Sep 10 2024

Keywords

Examples

			Consider the set S = {1, 2, 3}. The a(3) = 1 + 5 + 3 = 9 in the poset of permutations of {1,2,3}:
 |{(1)(2)(3)}| = 1;
 |{(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123), (1)(2)(3) < (132)}|=5;
 |{(1)(2)(3) < (1)(23) < (123), (1)(2)(3) < (2)(13)< (132), (1)(2)(3) < (3)(12) < (123)}| = 3.

Links

Crossrefs

Row sums of A375837.
Cf. A048994, A331955, A330804, A331956, A331957, A375835, A375836.
Cf. A385521.

Programs

  • Maple
    a:= proc(n) option remember;
      1+add(abs(Stirling1(n, k))*a(k), k=1..n-1)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 01 2025
  • Mathematica
    T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[(n == 0 && k == 0) || k == 1, 1, Sum[If[r >= 0, Abs[StirlingS1[n, r]]*T[r, k - 1], 0], {r, k - 1, n - 1}]]]; Table[Sum[T[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 01 2025, after A375837 *)

Formula

a(n) = Sum_{k=0..n} A375837(n,k).
a(n) = (A375836(n)+1)/2.
Conjecture: a(n) = R(n,0) where R(n,k) = (k+1) * (Sum_{i=0..n-1} R(n-1,i) + Sum_{j=0..k-1} R(n-1,j)) for 0 <= k < n, R(n,n) = 1. - Mikhail Kurkov, Jun 21 2025
a(n) ~ c * n!^2 / (2^n * log(2)^n * n^(1-log(2)/3)), where c = A385521 = 1.59585433050036621247006569740016516964502505848324064247941890934119103861277... - Vaclav Kotesovec, Jul 01 2025
a(n) = 1 + Sum_{k=1..n-1} abs(Stirling1(n,k))*a(k). - Rajesh Kumar Mohapatra, Jul 01 2025

A375835 Triangle read by rows: T(n, k) is the number of chains of length k in the poset of permutations of an n-set.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 8, 3, 0, 24, 64, 59, 18, 0, 120, 574, 970, 695, 180, 0, 720, 5858, 16124, 20240, 11955, 2700, 0, 5040, 67752, 285264, 556591, 559895, 282555, 56700, 0, 40320, 880584, 5459712, 15519287, 23585870, 19879370, 8780940, 1587600, 0, 362880, 12746208, 113511982, 451541898, 971214825, 1213062690, 882179550, 347072040, 57153600
Offset: 0

Views

Author

Rajesh Kumar Mohapatra, Aug 31 2024

Keywords

Examples

			The triangle T(n,k) begins:
  n\k 0    1      2      3      4       5      6      7 ...
  0   1
  1   0    1
  2   0    2      1
  3   0    6      8      3
  4   0   24     64     59     18
  5   0  120    574    970    695     180
  6   0  720   5858  16124  20240   11955   2700
  7   0 5040  67752 285264 556591  559895 282555  56700
  ...
The T(3, 2) = 8 chains in the poset of the permutations of {1, 2, 3} are:
{(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123),(1)(2)(3) < (132), (1)(23) < (123), (2)(13) < (132), (3)(12) < (123)}.

Crossrefs

Cf. A000007 (column k=0), A000142 (column k=1), A006472 (main diagonal), A375836 (row sums).
Cf. A048994, A331955, A330804, A331956, A331957.

Programs

  • Maple
    b := proc(n, k, t) option remember; if k < 0 then return 0 fi; if {n, k} = {0} then return 1 fi; add(ifelse(k = 1, 1, b(v, k - 1, 1))*abs(Stirling1(n, v)), v = k..n-t) end: T := (n, k) -> b(n, k, 0): seq((seq(T(n, k), k=0..n)), n = 0..10);  # Peter Luschny, Sep 05 2024
  • Mathematica
    b[n_, k_, t_] := b[n, k, t] = If[k < 0, 0, If[n == 0 && k == 0, 1,
Sum[If[k == 1, 1, b[v, k - 1, 1]] * Abs[StirlingS1[n, v]], {v, k, n - t}]]];
T[n_, k_] := b[n, k, 0]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]

Formula

Let Stirling1(n, k) denote the unsigned Stirling numbers of the first kind (A132393).
T(0, 0) = 1, T(0, k) = 0 for k > 0.
T(n, k) = Sum_{i_k=k..n} (Sum_{i_(k-1)=k-1..i_k - 1} (... (Sum_{i_2=2..i_3 - 1} (Sum_{i_1=1..i_2 - 1} Stirling1(n, i_k) * Stirling1(i_k, i_(k-1)) * ... * Stirling1(i_3, i_2) * Stirling1(i_2, i_1)))...)), where 1 <= k <= n.

A375837 Triangle read by rows: T(n,k) is the number of rooted chains starting with the cycle (1)(2)(3)...(n) of length k of permutation poset of n letters.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 5, 3, 0, 1, 23, 41, 18, 0, 1, 119, 455, 515, 180, 0, 1, 719, 5139, 10985, 9255, 2700, 0, 1, 5039, 62713, 222551, 334040, 225855, 56700, 0, 1, 40319, 840265, 4619447, 10899840, 12686030, 7193340, 1587600, 0, 1, 362879, 12383329, 101128653, 350413245, 620801580, 592261110, 289918440, 57153600
Offset: 0

Views

Author

Rajesh Kumar Mohapatra, Ranjan Kumar Dhani, and Subhashree Sahoo, Aug 31 2024

Keywords

Examples

			Triangle T(n,k) begins:
  n\k | 0  1   2     3     4      5      6     7 ...
 -----+-----------------------------------------
  0   | 1;
  1   | 0, 1;
  2   | 0, 1, 1;
  3   | 0, 1, 5, 3;
  4   | 0, 1, 23, 41, 18;
  5   | 0, 1, 119, 455, 515, 180;
  6   | 0, 1, 719, 5139, 10985, 9255, 2700;
  7   | 0, 1, 5039, 62713, 222551, 334040, 225855, 56700;
  ...
The T(3, 2) = 5 chains in the poset of the permutations of {1, 2, 3} are: {(1)(2)(3) < (1)(23), (1)(2)(3) < (2)(13), (1)(2)(3) < (3)(12), (1)(2)(3) < (123),(1)(2)(3) < (132)}.

Links

Crossrefs

Cf. A000007 (column k=0), A057427 (column k=1), A006472 (diagonal), A375838 (row sums).
Cf. A048994, A331955, A330804, A331956, A331957, A375835, A375836.

Programs

  • Mathematica
    T[n_, k_] := T[n, k] = If[k < 0 || k > n, 0, If[(n == 0 && k == 0) || k == 1, 1, Sum[If[r >= 0, Abs[StirlingS1[n, r]]*T[r, k - 1], 0], {r, k - 1, n - 1}]]]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* corrected Jul 01 2025 *)

Formula

Let Stirling1(n, k) denote the unsigned Stirling numbers of the first kind (A132393).
T(0, 0) = 1, T(0, k) = 0 for k > 0 and T(n, 1) = 1 for n > 1.
T(n, k) = Sum_{i_(k-1)=k-1..n-1} (Sum_{i_(k-2)=k-2..i_(k-1) - 1} (... (Sum_{i_2=2..i_3 - 1} (Sum_{i_1=1..i_2 - 1} Stirling1(n,i_(k-1)) * Stirling1(i_(k-1),i_(k-2)) * ... * Stirling1(i_3,i_2) * Stirling1(i_2,i_1)))...)), where 2 <= k <= n.
Showing 1-3 of 3 results.

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