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.

A055887 Number of ordered partitions of partitions.

Original entry on oeis.org

1, 1, 3, 8, 22, 59, 160, 431, 1164, 3140, 8474, 22864, 61697, 166476, 449210, 1212113, 3270684, 8825376, 23813776, 64257396, 173387612, 467856828, 1262431711, 3406456212, 9191739970, 24802339472, 66924874539, 180585336876, 487278670744, 1314838220172
Offset: 0

Views

Author

Christian G. Bower, Jun 09 2000

Keywords

Comments

Jordan matrices are upper bidiagonal matrices such that (A) the diagonal entries are in sorted order, (B) there are only 1's and 0's on the superdiagonal, (C) for each superdiagonal 1, the two diagonal entries to the left and below it must be equal. Let J(N) be the number of N X N Jordan matrices where the diagonal values are, without loss of generality, taken to be a prefix of some fixed strictly increasing sequence x_1, x_2, x_3, ... If Jordan blocks sorted by eigenvalue with ties broken by block size during the sorting, then J(1, 2, 3, ...) is this sequence. - Warren D. Smith, Jan 28 2002
Number of compositions of n into parts k >= 1 where there are A000041(k) sorts of part k. - Joerg Arndt, Sep 30 2012
Also number of chains of multisets that partition a normal multiset of weight n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Oct 28 2015
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into constant multisets of a multiset of length n covering an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(3) = 8 multiset partitions are:
{{1}} {{1,1}} {{1,1,1}}
{{1},{1}} {{1},{1,1}}
{{1},{2}} {{1},{2,2}}
{{2},{1,1}}
{{1},{1},{1}}
{{1},{1},{2}}
{{1},{2},{2}}
{{1},{2},{3}}
Factorizations into prime powers, are counted by A000688.
The strongly normal case is A063834.
The strongly normal strict case is A270995.
Twice-partitions of type PPR are counted by A279784, factorizations A295935.
The strict case is A304969.
(End)

Examples

			The a(4) = 22 chains of multisets, where notation x-y means "y is a submultiset of x", are: (o-o-o-o) (oo-o-o) (oo-oo) (ooo-o) (oooo) (oe-o-o) (ooe-o) (oooe) (oe-oe) (ooe-e) (oee-o) (ooee) (oei-o) (ooei) (oe-e-e) (oee-e) (oeee) (oei-e) (oeei) (oei-i) (oeii) (oeis).
From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose an integer partition of each part of an integer composition of n. The a(0) = 1 through a(3) = 8 choices are:
  ()  ((1))  ((2))     ((3))
             ((11))    ((21))
             ((1)(1))  ((111))
                       ((1)(2))
                       ((2)(1))
                       ((1)(11))
                       ((11)(1))
                       ((1)(1)(1))
(End)

Links

Crossrefs

Cf. A055888, A083355, A095975, A246828.
Row sums of A060642.
Cf. A326346.
The unordered version is A001970, row-sums of A061260.
A000041 counts integer partitions, strict A000009.
A011782 counts integer compositions.
A072233 counts partitions by sum and length.
Cf. A001055, A050361, A063834, A279784, A355743.

Programs

  • Maple
    with(combstruct); SeqSetSetU := [T, {T=Sequence(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
P := (x) -> product( 1/(1-x^k), k=1..20 ) - 1; F := (x) -> series( 1/(1-P(x)) - 1, x, 21 ); # F(x) is g.f. for this sequence # Warren D. Smith, Jan 28 2002
A055887rec:= proc(n::integer) local k; option remember; with(combinat): if n = 0 then 1 else add(numbpart(k) *procname(n - k), k=1..n); end if; end proc: seq (A055887rec(n), n=0..10); # Thomas Wieder, Nov 26 2007
  • Mathematica
    a = 1/Product[(1 - x^k), {k, 1, \[Infinity]}] - 1; CoefficientList[Series[1/(1 - a), {x, 0, 20}], x] (* Geoffrey Critzer, Dec 23 2010 *)
(1/(2 - 1/QPochhammer[x]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
Table[Sum[Times@@PartitionsP/@c,{c,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}] (* Gus Wiseman, Jul 31 2022 *)
  • PARI
    Vec(1/(2-1/eta(x+O(x^66)))) \\ Joerg Arndt, Sep 30 2012

Formula

Invert transform of partitions numbers A000041.
Let p(k) be the number of integer partitions of k. Furthermore, set a(0)=1. Then a(n) = Sum_{k=1..n} p(k)*a(n-k). - Thomas Wieder, Nov 26 2007
G.f.: 1/( 1 - Sum_{k>=1} p(k)*x^k ) where p(k) = A000041(k) is the number of integer partitions of k. - Joerg Arndt, Sep 30 2012
a(n) ~ c * d^n, where d = 2.698329106474211231263998666188376330713465125913986356769... (see A246828) and c = 0.414113793172792357745578049739573823627306487211379286647... - Vaclav Kotesovec, Mar 29 2014
G.f.: 1/(2 - 1/(x)inf), where (x)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Sep 22 2016

A060642 Triangle read by rows: row n lists number of ordered partitions into k parts of partitions of n.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 22, 21, 8, 1, 11, 43, 59, 36, 10, 1, 15, 80, 144, 124, 55, 12, 1, 22, 141, 321, 362, 225, 78, 14, 1, 30, 240, 669, 944, 765, 370, 105, 16, 1, 42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1, 56, 640, 2511, 5100, 6215, 4848, 2478, 824, 171, 20, 1
Offset: 1

Views

Author

Alford Arnold, Apr 16 2001

Keywords

Comments

Also the convolution triangle of A000041. - Peter Luschny, Oct 07 2022

Examples

			Table begins:
   1;
   2,   1;
   3,   4,    1;
   5,  10,    6,    1;
   7,  22,   21,    8,    1;
  11,  43,   59,   36,   10,    1;
  15,  80,  144,  124,   55,   12,   1;
  22, 141,  321,  362,  225,   78,  14,   1;
  30, 240,  669,  944,  765,  370, 105,  16,  1;
  42, 397, 1323, 2266, 2287, 1437, 567, 136, 18, 1;
  ...
For n=4 there are 5 partitions of 4, namely 4, 31, 22, 211, 11111. There are 5 ways to pick 1 of them; 10 ways to partition one of them into 2 ordered parts: 3,1; 1,3; 2,2; 21,1; 1,21; 2,11; 11,2; 111,1; 1,111; 11,11; 6 ways to partition one of them into 3 ordered parts: 2,1,1; 1,2,1; 1,1,2; 11,1,1; 1,11,1; 1,1,11; and one way to partition one of them into 4 ordered parts: 1,1,1,1. So row 4 is 5,10,6,1.

Links

Crossrefs

Columns k=1-10 give: A000041, A048574, A341221, A341222, A341223, A341225, A341226, A341227, A341228, A341236.
Row sums give A055887.
Cf. A097805, A010815, A055888, A144064, A261719, A326346.
T(2n,n) gives A340987.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, k*add(
      A(n-j, k)*numtheory[sigma](j), j=1..n)/n)
    end:
T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 12 2015
# Uses function PMatrix from A357368. Adds row and column for n, k = 0.
PMatrix(10, combinat:-numbpart); # Peter Luschny, Oct 07 2022
  • Mathematica
    A[n_, k_] := A[n, k] = If[n==0, 1, k*Sum[A[n-j, k]*DivisorSigma[1, j], {j, 1, n}]/n]; T[n_, k_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[ Table[ T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

Formula

G.f. A(n;x) for n-th row satisfies A(n;x) = Sum_{k=0..n-1} A000041(n-k)*A(k;x)*x, A(0;x) = 1. - Vladeta Jovovic, Jan 02 2004
T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A144064(n,k-i). - Alois P. Heinz, Mar 12 2015
Sum_{k=1..n} k * T(n,k) = A326346(n). - Alois P. Heinz, Sep 11 2019
Sum_{k=0..n} (-1)^k * T(n,k) = A010815(n). - Alois P. Heinz, Feb 07 2021
G.f. of column k: (-1 + Product_{j>=1} 1 / (1 - x^j))^k. - Ilya Gutkovskiy, Feb 13 2021

Extensions

More terms from Vladeta Jovovic, Jan 02 2004

A246828 Decimal expansion of a constant related to A055887.

Original entry on oeis.org

2, 6, 9, 8, 3, 2, 9, 1, 0, 6, 4, 7, 4, 2, 1, 1, 2, 3, 1, 2, 6, 3, 9, 9, 8, 6, 6, 6, 1, 8, 8, 3, 7, 6, 3, 3, 0, 7, 1, 3, 4, 6, 5, 1, 2, 5, 9, 1, 3, 9, 8, 6, 3, 5, 6, 7, 6, 9, 0, 1, 2, 3, 1, 1, 7, 8, 1, 9, 8, 6, 5, 9, 3, 6, 6, 9, 5, 0, 5, 5, 9, 4, 5, 1, 3, 6, 6, 4, 7, 6, 6, 5, 2, 0, 2, 2, 0, 3, 5, 5, 8, 0, 0, 7, 7
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 04 2014

Keywords

Examples

			2.698329106474211231263998666188376330713465125913986356769...

Links

Crossrefs

Cf. A055887, A141799, A095975, A131408, A326346.

Programs

  • Mathematica
    RealDigits[1/x /. FindRoot[QPochhammer[x] == 1/2, {x, 1/2}, WorkingPrecision -> 120]][[1]] (* Vaclav Kotesovec, May 23 2018 *)

Formula

Equals lim n -> infinity A055887(n)^(1/n).
Equals lim n -> infinity A095975(n)^(1/n).
Equals lim n -> infinity A141799(n)^(1/n).
Equals lim n -> infinity A131408(n)^(1/n).
Root of the equation QPochhammer[x] = 1/2. - Vaclav Kotesovec, May 23 2018

A327548 Total number of compositions in the compositions of partitions of n.

Original entry on oeis.org

0, 1, 4, 11, 34, 85, 248, 603, 1630, 4017, 10308, 24855, 63210, 150141, 369936, 882083, 2135606, 5023689, 12064092, 28167919, 66828418, 155569685, 364983208, 844175675, 1971322574, 4533662817, 10498550260, 24077361031, 55432615194, 126492183213, 289997946944
Offset: 0

Views

Author

Alois P. Heinz, Sep 16 2019

Keywords

Examples

			a(3) = 11 = 1+1+1+1+2+2+3 counts the compositions in 3, 21, 12, 111, 2|1, 11|1, 1|1|1.

Links

Crossrefs

Cf. A000041, A011782, A075900, A326346, A327549.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
      b(n, i-1)+(p->p+[0, p[1]])(2^(i-1)*b(n-i, min(n-i, i)))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..32);
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, b[n, i - 1] + With[{p = 2^(i - 1) b[n - i, Min[n - i, i]]}, p + {0, p[[1]]}]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 32] (* Jean-François Alcover, Dec 17 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A327549(n,k).
a(n) ~ log(2) * (3/(Pi^2 - 6*log(2)^2))^(1/4) * 2^(n-1) * exp(sqrt((Pi^2 - 6*log(2)^2)*n/3)) / (sqrt(Pi) * n^(1/4)). - Vaclav Kotesovec, Sep 19 2019
Showing 1-4 of 4 results.

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