A346772 -id:A346772 - cp's OEIS frontend

A270236 Triangle T(n,p) read by rows: the number of occurrences of p in the restricted growth functions of length n.

1, 3, 1, 9, 5, 1, 30, 21, 8, 1, 112, 88, 47, 12, 1, 463, 387, 253, 97, 17, 1, 2095, 1816, 1345, 675, 184, 23, 1, 10279, 9123, 7304, 4418, 1641, 324, 30, 1, 54267, 48971, 41193, 28396, 13276, 3645, 536, 38, 1, 306298, 279855, 243152, 183615, 102244, 36223, 7473, 842, 47, 1

Offset: 1

R. J. Mathar, Mar 13 2016

The RG functions used here are defined by f(1)=1, f(j) <= 1+max_{i

T(n,p) is the number of elements in the p-th subset in all set partitions of [n]. - Joerg Arndt, Mar 14 2016

			The two restricted growth functions of length 2 are (1,1) and (1,2). The 1 appears 3 times and the 2 once, so T(2,1)=3 and T(2,2)=1.
1;
3,1;
9,5,1;
30,21,8,1;
112,88,47,12,1;
463,387,253,97,17,1;
2095,1816,1345,675,184,23,1;
10279,9123,7304,4418,1641,324,30,1;
54267,48971,41193,28396,13276,3645,536,38,1;
306298,279855,243152,183615,102244,36223,7473,842,47,1;
1838320,1695902,1506521,1211936,770989,334751,90223,14303,1267,57,1;
11677867,10856879,9799547,8237223,5795889,2965654,995191,207186,25820, 1839,68,1;

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A070071 (row sums).
Column p=1-10 gives: A124427, A270494, A270495, A270496, A270497, A270498, A270499, A270500, A270501, A270502.
T(2n+1,n+1) gives A270529.
Cf. A000110, A185105, A285362, A286416, A346772.

b:= proc(n, m) option remember; `if`(n=0, [1, 0], add((p->
      [p[1], p[2]+p[1]*x^j])(b(n-1, max(m, j))), j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n, 0)[2]):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 14 2016

b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]] + p[[1]]*x^j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, 0][[2]] ]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

T(n,n) = 1.
Conjecture: T(n,n-1) = 2+n*(n-1)/2 for n>1.
Conjecture: T(n+1,n-1) = 2+n*(n+1)*(3*n^2-5*n+26)/24 for n>1.
Sum_{k=1..n} k * T(n,k) = A346772(n). - Alois P. Heinz, Aug 03 2021

A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)B(k,q)q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 4, 2, 1, 1, 1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1, 1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1, 1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1, 1, 7, 21, 56, 105, 161, 231, 302, 356, 379, 392, 384, 358, 314, 262

Offset: 0

Paul D. Hanna, Dec 31 2006, May 28 2007

Limit of reversed rows equals A126348. Largest term in rows equal A126349.

			Number of terms in row n is: n*(n-1)/2 + 1.
Row functions B(n,q) begin:
  B(0,q) = 1;
  B(1,q) = 1;
  B(2,q) = 1 + q;
  B(3,q) = 1 + 2*q + q^2 + q^3;
  B(4,q) = 1 + 3*q + 3*q^2 + 4*q^3 + 2*q^4 + q^5 + q^6.
Triangle begins:
  1;
  1;
  1, 1;
  1, 2, 1, 1;
  1, 3, 3, 4, 2, 1, 1;
  1, 4, 6, 10, 9, 7, 7, 4, 2, 1, 1;
  1, 5, 10, 20, 25, 26, 29, 26, 20, 14, 12, 7, 4, 2, 1, 1;
  1, 6, 15, 35, 55, 71, 90, 101, 100, 89, 82, 68, 53, 38, 26, 20, 12, 7, 4, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..60, flattened
Carl G. Wagner, Partition Statistics and q-Bell Numbers (q = -1), J. Integer Seqs., Vol. 7, 2004.

Row sums give A000110.
Cf. A126348, A126349; factorial variant: A126470.
Cf. A346772.

b:= proc(n, m, t) option remember; `if`(n=0, x^t,
      add(b(n-1, max(m, j), t+j) , j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=n..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Aug 02 2021

B[0, ] = 1; B[n, q_] := B[n, q] = Sum[Binomial[n-1, k] B[k, q] q^k, {k, 0, n-1}] // Expand; Table[CoefficientList[B[n, q], q], {n, 0, 8}] // Flatten (* Jean-François Alcover, Nov 08 2016 *)

{B(n,q)=if(n==0,1,sum(k=0,n-1,binomial(n-1,k)*B(k,q)*q^k))}
row(n)={Vec(B(n, 'q)+O('q^(n*(n-1)/2+1)))}

/* Alternative formula for the n-th q-Bell number (row n): */ {B(n,q)=local(inf=100);round((0^n + sum(k=1, inf,((q^k-1)/(q-1))^n/prod(i=1,k,(q^i-1)/(q-1)))) / prod(k=1, inf,1 + (q-1)/q^k))}

G.f. for row n: B(n,q) = 1/E_q*{0^n + Sum_{k>=1} [(q^k-1)/(q-1)]^n / q-Factorial(k)}, where q-Factorial(k) = Product_{j=1..k} [(q^j-1)/(q-1)] and where E_q = Sum_{n>=0} 1/q-Factorial(n) = Product_{n>=1} (1+(q-1)/q^n).
Sum_{k=0..n*(n-1)/2} (n+k) * T(n,k) = A346772(n). - Alois P. Heinz, Aug 02 2021
Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - Mikhail Kurkov, Jul 06 2025

Keyword:tabl changed to tabf by R. J. Mathar, Oct 21 2010

A120057 Table T(n,k) = sum over all set partitions of n of number at index k.

Original entry on oeis.org

1, 2, 3, 5, 8, 9, 15, 25, 29, 31, 52, 89, 106, 115, 120, 203, 354, 431, 474, 499, 514, 877, 1551, 1923, 2141, 2273, 2355, 2407, 4140, 7403, 9318, 10489, 11224, 11695, 12002, 12205, 21147, 38154, 48635, 55286, 59595, 62434, 64331, 65614, 66491, 115975, 210803, 271617, 311469, 338019, 355951, 368205, 376665, 382559, 386699

Offset: 1

Franklin T. Adams-Watters, Jun 06 2006, Jun 07 2006

			The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}. Summing these componentwise gives the third row: 5,8,9.
Table starts:
   1;
   2,  3;
   5,  8,   9;
  15, 25,  29,  31;
  52, 89, 106, 115, 120;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A120058, A120095. First column is A000110.
Main diagonal is A087648(n-1).
Row sums give A346772.

b:= proc(n, m) option remember; `if`(n=0, [1, 0],
      add((p-> [p[1], expand(p[2]*x+p[1]*j)])(
        b(n-1, max(m, j))), j=1..m+1))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n-1))(b(n, 0)[2]):
seq(T(n), n=1..10);  # Alois P. Heinz, Mar 24 2016

b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[Function[p, {p[[1]], p[[2]]*x + p[[1]]*j}][b[n-1, Max[m, j]]], {j, 1, m+1}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n-1}]][b[n, 0][[2]]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Apr 08 2016, after Alois P. Heinz *)

T(n,k) = Sum_{i=1..k} A120058(n,i)*B(n-i+1), where B(n) are the Bell numbers, (A000110).

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cp's OEIS Frontend

A270236 Triangle T(n,p) read by rows: the number of occurrences of p in the restricted growth functions of length n.

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A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)B(k,q)q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

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A120057 Table T(n,k) = sum over all set partitions of n of number at index k.

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cp's OEIS Frontend

A270236 Triangle T(n,p) read by rows: the number of occurrences of p in the restricted growth functions of length n.

Views

Author

Keywords

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Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A120057 Table T(n,k) = sum over all set partitions of n of number at index k.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A126347 Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)B(k,q)q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).