A276925 -id:A276925 - cp's OEIS frontend

A276921 Number A(n,k) of ordered set partitions of [n] with at most k elements per block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 6, 0, 1, 1, 3, 12, 24, 0, 1, 1, 3, 13, 66, 120, 0, 1, 1, 3, 13, 74, 450, 720, 0, 1, 1, 3, 13, 75, 530, 3690, 5040, 0, 1, 1, 3, 13, 75, 540, 4550, 35280, 40320, 0, 1, 1, 3, 13, 75, 541, 4670, 45570, 385560, 362880, 0

Offset: 0

Alois P. Heinz, Sep 22 2016

			Square array A(n,k) begins:
  1,    1,     1,     1,     1,     1,     1,     1, ...
  0,    1,     1,     1,     1,     1,     1,     1, ...
  0,    2,     3,     3,     3,     3,     3,     3, ...
  0,    6,    12,    13,    13,    13,    13,    13, ...
  0,   24,    66,    74,    75,    75,    75,    75, ...
  0,  120,   450,   530,   540,   541,   541,   541, ...
  0,  720,  3690,  4550,  4670,  4682,  4683,  4683, ...
  0, 5040, 35280, 45570, 47110, 47278, 47292, 47293, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.

Columns k=0..10 give: A000007, A000142, A080599, A189886, A276924, A276925, A276926, A276927, A276928, A276929, A276930.
Main diagonal gives A000670.
Cf. A276922.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
       A(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[A[n-i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 03 2017, translated from Maple *)

E.g.f. of column k: 1/(1-Sum_{i=1..k} x^i/i!).

A(n,k) = Sum_{j=0..k} A276922(n,j).

A110038 The partition function G(n,5).

Original entry on oeis.org

1, 1, 2, 5, 15, 52, 202, 869, 4075, 20645, 112124, 648649, 3976633, 25719630, 174839120, 1245131903, 9263053753, 71806323461, 578719497070, 4839515883625, 41916097982471, 375401824277096, 3471395994487422, 33099042344383885, 325005134436155395

Offset: 0

N. J. A. Sloane, May 13 2009

nonn

Set partitions into sets of size at most 5. The e.g.f. for partitions into sets of size at most s is exp( sum(j=1..s, x^j/j!) ). [Joerg Arndt, Dec 07 2012]

Seiichi Manyama, Table of n, a(n) for n = 0..591 (terms 0..200 from Alois P. Heinz)
Moa Apagodu, David Applegate, N. J. A. Sloane, and Doron Zeilberger, Analysis of the Gift Exchange Problem, arXiv:1701.08394 [math.CO], 2017.
David Applegate and N. J. A. Sloane, The Gift Exchange Problem, arXiv:0907.0513 [math.CO], 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
F. L. Miksa, L. Moser and M. Wyman, Restricted partitions of finite sets, Canad. Math. Bull., 1 (1958), 87-96.

The sequences G(n,1), G(n,2), G(n,3), G(n,4), G(n,5), G(n,6) are given by A000012, A000085, A001680, A001681, A110038, A148092 respectively.
Column k=5 of A229223.
Cf. A276925.

G:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(G(n-i*j, i-1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
a:= n-> G(n, 5):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 20 2012
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [1, 1, 2, 5, 15][n+1],
      a(n-1)+(n-1)*(a(n-2)+(n-2)/2*(a(n-3)+(n-3)/3*(a(n-4)
      +(n-4)/4*a(n-5)))))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 15 2013

G[n_, i_] := G[n, i] = If[n == 0, 1, If[i<1, 0, Sum[G[n-i*j, i-1] *n!/i!^j/(n-i*j)!/j!, {j, 0, n/i}]]]; a[n_] := G[n, 5]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)

E.g.f.: exp( x + x^2/2 + x^3/6 + x^4/24 + x^5/120 ).
a(n) = n! * sum(k=1..n, 1/k! * sum(r=0..k, C(k,r) * sum(m=0..r, 2^(m-r) * C(r,m) * sum(j=0..m, C(m,j) * C(j,n-m-k-j-r) * 6^(j-m) * 24^(n-r-m-k-2*j) * 120^(m+k+j+r-n))))). - Vladimir Kruchinin, Jan 25 2011
a(n) = G(n,5) with G(0,i) = 1, G(n,i) = 0 for n>0 and i<1, otherwise G(n,i) = Sum_{j=0..floor(n/i)} G(n-i*j,i-1) * n!/(i!^j*(n-i*j)!*j!). - Alois P. Heinz, Apr 20 2012

A320761 Number of ordered set partitions of [n] where the maximal block size equals five.

Original entry on oeis.org

1, 12, 168, 2464, 38808, 657972, 11997216, 234594360, 4903616718, 109205019924, 2582909885556, 64686057980544, 1710536977653504, 47637803779229664, 1393903719674129664, 42758329987344875904, 1372254504736418142840, 45989719374155059863360

Offset: 5

Alois P. Heinz, Oct 20 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..424

Column k=5 of A276922.

Cf. A276924, A276925.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
a:= n-> (k-> b(n, k) -b(n, k-1))(5):
seq(a(n), n=5..25);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - i, k] Binomial[n, i], {i, 1, Min[n, k]}]];
a[n_] := With[{k = 5}, b[n, k] - b[n, k-1]];
a /@ Range[5, 25] (* Jean-François Alcover, Dec 14 2020, after Alois P. Heinz *)

E.g.f.: 1/(1-Sum_{i=1..5} x^i/i!) - 1/(1-Sum_{i=1..4} x^i/i!).

a(n) = A276925(n) - A276924(n).

A320762 Number of ordered set partitions of [n] where the maximal block size equals six.

Original entry on oeis.org

1, 14, 224, 3696, 64680, 1206744, 24011988, 508864356, 11459682234, 273563089800, 6904861371408, 183819838041840, 5149305370473264, 151447801937560128, 4666989813249123360, 150394712706368466336, 5059062167993588722968, 177346570951333803395376

Offset: 6

Alois P. Heinz, Oct 20 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..424

Column k=6 of A276922.

Cf. A276925, A276926.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
a:= n-> (k-> b(n, k) -b(n, k-1))(6):
seq(a(n), n=6..25);

E.g.f.: 1/(1-Sum_{i=1..6} x^i/i!) - 1/(1-Sum_{i=1..5} x^i/i!).

a(n) = A276926(n) - A276925(n).

cp's OEIS Frontend

A276921 Number A(n,k) of ordered set partitions of [n] with at most k elements per block; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A110038 The partition function G(n,5).

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A320761 Number of ordered set partitions of [n] where the maximal block size equals five.

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Maple

Mathematica

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A320762 Number of ordered set partitions of [n] where the maximal block size equals six.

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Maple

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