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 A355384 #14 May 09 2025 01:45:34
%S A355384 1,1,2,4,12,30,66,164,419,1049,2625,6372,15451,37335,89855,216523,
%T A355384 518714,1235897,2930050,6911149,16217817,37914515,88304358,204971388,
%U A355384 474172899,1093547574,2513959446,5761735383,13165908506,29998936859,68164839887,154478212575
%N A355384 Number of pairs (y, v) where y is a composition of n and v is a (not necessarily contiguous) subsequence of y whose length equals the number of distinct parts in y.
%C A355384 If a composition is regarded as an arrow from the number of parts to the number of distinct parts, this sequence counts composable containments of compositions.
%H A355384 Christian Sievers, <a href="/A355384/b355384.txt">Table of n, a(n) for n = 0..59</a>
%e A355384 The initial terms count the following containments:
%e A355384 ()() (1)(1) (2)(2) (3)(3) (4)(4)
%e A355384 (11)(1) (21)(21) (31)(31)
%e A355384 (12)(12) (13)(13)
%e A355384 (111)(1) (22)(2)
%e A355384 (211)(11)
%e A355384 (211)(21)
%e A355384 (121)(11)
%e A355384 (121)(12)
%e A355384 (121)(21)
%e A355384 (112)(11)
%e A355384 (112)(12)
%e A355384 (1111)(1)
%t A355384 Table[Sum[Length[Union[Subsets[y,{Length[Union[y]]}]]],{y,Join@@Permutations/@IntegerPartitions[n]}],{n,0,5}]
%Y A355384 The homog. case is A032020, w/o containment A355388 (partitions A355385).
%Y A355384 For partitions we have A355383, homog. A000009, w/ multiplicity A339006.
%Y A355384 A000244 counts splittings of compositions, for partitions A323583.
%Y A355384 Cf. A001970, A022811, A063834, A070933, A072706, A133494, A336139.
%K A355384 nonn
%O A355384 0,3
%A A355384 _Gus Wiseman_, Jul 01 2022
%E A355384 a(21) and beyond from _Christian Sievers_, May 08 2025