A281912 -id:A281912 - cp's OEIS frontend

A338169 A list of all finite sequences of distinct positive integers sorted by their maximum value, then lexicographically.

1, 1, 2, 2, 2, 1, 1, 2, 3, 1, 3, 1, 3, 2, 2, 1, 3, 2, 3, 2, 3, 1, 3, 3, 1, 3, 1, 2, 3, 2, 3, 2, 1, 1, 2, 3, 4, 1, 2, 4, 1, 2, 4, 3, 1, 3, 2, 4, 1, 3, 4, 1, 3, 4, 2, 1, 4, 1, 4, 2, 1, 4, 2, 3, 1, 4, 3, 1, 4, 3, 2, 2, 1, 3, 4, 2, 1, 4, 2, 1, 4, 3, 2, 3, 1, 4, 2

Offset: 1

Peter Kagey, Oct 14 2020

There are A001339(n - 1) sequences with a maximum value of n.

Conjecture: the sum of the sum of all sequences with a maximum value of n is given by A281912(n).

			Maximum value 1:
[1],
Maximum value 2:
[1,2],
[2],
[2,1],
Maximum value 3:
[1,2,3],
[1,3],
[1,3,2],
[2,1,3],
[2,3],
[2,3,1],
[3],
[3,1],
[3,1,2],
[3,2],
[3,2,1]

Peter Kagey, Table of n, a(n) for n = 1..9786 (all sequences with maximum <= 6.)

Cf. A001339, A026791, A228369, A281912.

Table[Map[ToExpression /@ StringPartition[#, 1, 1] &, Sort[Map[StringJoin, Join @@ Map[Permutations, DeleteCases[Rest@ Subsets[ToString /@ Range@ k], ?(FreeQ[#, ToString@ k] &)]]]]], {k, 4}] // Flatten (* _Michael De Vlieger, Oct 15 2020 *)

A292998 Number of sequences of balls colored with at most n colors such that exactly three balls are the same color as some other ball in the sequence.

Original entry on oeis.org

1, 10, 87, 772, 7285, 74046, 812875, 9626632, 122643657, 1675253170, 24449818591, 379984902540, 6268557335677, 109443030279142, 2016658652491155, 39119860206021136, 797013832285599505, 17017679492994949722, 380045072079456330727

Offset: 1

Jeremy Dover, Sep 27 2017

nonn

Note that any such sequence has at least 3 balls and at most n+2, and that three matching balls must all be the same color.

			For n=2 colors a, b, the a(n)=10 sequences of balls are: aaa, bbb, abbb, babb, bbab, bbba, baaa, abaa, aaba, aaab.

Jeremy M. Dover, Repetition in Colored Sequences of Balls, arXiv:1710.06049 [math.CO], 2017.

Row sums of triangle A292930.

Cf. A281912, A292999.

Table[n!*Sum[Binomial[k, 3]/(n + 2 - k)!, {k, 3, n + 2}], {n, 19}] (* Michael De Vlieger, Sep 28 2017 *)

a(n) = n! * sum(k=3, n+2, binomial(k,3)/(n+2-k)!); \\ Michel Marcus, Sep 29 2017

a(n) = n! * Sum_{k=3..n+2} binomial(k,3)/(n+2-k)!.

cp's OEIS Frontend

A338169 A list of all finite sequences of distinct positive integers sorted by their maximum value, then lexicographically.

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