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.
%I A055887 #74 Feb 16 2025 08:32:43
%S A055887 1,1,3,8,22,59,160,431,1164,3140,8474,22864,61697,166476,449210,
%T A055887 1212113,3270684,8825376,23813776,64257396,173387612,467856828,
%U A055887 1262431711,3406456212,9191739970,24802339472,66924874539,180585336876,487278670744,1314838220172
%N A055887 Number of ordered partitions of partitions.
%C A055887 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
%C A055887 Number of compositions of n into parts k >= 1 where there are A000041(k) sorts of part k. - _Joerg Arndt_, Sep 30 2012
%C A055887 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
%C A055887 From _Gus Wiseman_, Jul 31 2022: (Start)
%C A055887 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:
%C A055887 {{1}} {{1,1}} {{1,1,1}}
%C A055887 {{1},{1}} {{1},{1,1}}
%C A055887 {{1},{2}} {{1},{2,2}}
%C A055887 {{2},{1,1}}
%C A055887 {{1},{1},{1}}
%C A055887 {{1},{1},{2}}
%C A055887 {{1},{2},{2}}
%C A055887 {{1},{2},{3}}
%C A055887 Factorizations into prime powers, are counted by A000688.
%C A055887 The strongly normal case is A063834.
%C A055887 The strongly normal strict case is A270995.
%C A055887 Twice-partitions of type PPR are counted by A279784, factorizations A295935.
%C A055887 The strict case is A304969.
%C A055887 (End)
%H A055887 Alois P. Heinz, <a href="/A055887/b055887.txt">Table of n, a(n) for n = 0..2320</a>
%H A055887 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A055887 N. J. A. Sloane and Thomas Wieder, <a href="https://arxiv.org/abs/math/0307064">The Number of Hierarchical Orderings</a>, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89, DOI:10.1007/s11083-004-9460-9.
%H A055887 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>.
%F A055887 Invert transform of partitions numbers A000041.
%F A055887 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
%F A055887 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
%F A055887 a(n) ~ c * d^n, where d = 2.698329106474211231263998666188376330713465125913986356769... (see A246828) and c = 0.414113793172792357745578049739573823627306487211379286647... - _Vaclav Kotesovec_, Mar 29 2014
%F A055887 G.f.: 1/(2 - 1/(x)_inf), where (x)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Sep 22 2016
%e A055887 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).
%e A055887 From _Gus Wiseman_, Jul 31 2022: (Start)
%e A055887 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:
%e A055887 () ((1)) ((2)) ((3))
%e A055887 ((11)) ((21))
%e A055887 ((1)(1)) ((111))
%e A055887 ((1)(2))
%e A055887 ((2)(1))
%e A055887 ((1)(11))
%e A055887 ((11)(1))
%e A055887 ((1)(1)(1))
%e A055887 (End)
%p A055887 with(combstruct); SeqSetSetU := [T, {T=Sequence(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
%p A055887 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
%p A055887 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
%t A055887 a = 1/Product[(1 - x^k), {k, 1, \[Infinity]}] - 1; CoefficientList[Series[1/(1 - a), {x, 0, 20}], x] (* _Geoffrey Critzer_, Dec 23 2010 *)
%t A055887 (1/(2 - 1/QPochhammer[x]) + O[x]^30)[[3]] (* _Vladimir Reshetnikov_, Sep 22 2016 *)
%t A055887 Table[Sum[Times@@PartitionsP/@c,{c,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}] (* _Gus Wiseman_, Jul 31 2022 *)
%o A055887 (PARI) Vec(1/(2-1/eta(x+O(x^66)))) \\ _Joerg Arndt_, Sep 30 2012
%Y A055887 Cf. A055888, A083355, A095975, A246828.
%Y A055887 Row sums of A060642.
%Y A055887 Cf. A326346.
%Y A055887 The unordered version is A001970, row-sums of A061260.
%Y A055887 A000041 counts integer partitions, strict A000009.
%Y A055887 A011782 counts integer compositions.
%Y A055887 A072233 counts partitions by sum and length.
%Y A055887 Cf. A001055, A050361, A063834, A279784, A355743.
%K A055887 nonn
%O A055887 0,3
%A A055887 _Christian G. Bower_, Jun 09 2000