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A268667 Number of sequences with j copies of j for each j in {1,2,...,n} and longest increasing subsequence of length n. 2
1, 1, 2, 26, 3511, 6742796, 233249911607, 175703195017370516, 3377940832101159287907151, 1899957346851645870857879683505890, 35246706696124014829643459097288501560957174, 23998872279553738609365779286317516184675391844037227392 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Sequences counted by a(n) have length A000217(n) and element sum A000330(n).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..15

J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

EXAMPLE

a(2) = 2: 122, 212.

a(3) = 26: 122333, 123233, 123323, 123332, 132233, 132323, 132332, 133223, 133232, 212333, 213233, 213323, 231233, 231323, 233123, 312233, 312323, 312332, 313223, 313232, 321233, 321323, 323123, 331223, 331232, 332123.

MAPLE

g:= proc(l) option remember; (n-> f(l[1..nops(l)-1])*

      binomial(n-1, l[-1]-1)+ add(f(sort(subsop(j=l[j]

      -1, l))), j=1..nops(l)-1))(add(i, i=l))

    end:

f:= l-> (n-> `if`(n<2 or l[-1]=1, 1, `if`(l[1]=0, 0, `if`(

         n=2, binomial(l[1]+l[2], l[1])-1, g(l)))))(nops(l)):

a:= n-> f([$1..n]):

seq(a(n), n=0..8);

MATHEMATICA

g[l_] := g[l] = Function[n, f[Most[l]]*Binomial[n-1, l[[-1]]-1] + Sum[f[ Sort[ ReplacePart[l, j -> l[[j]]-1]]], {j, 1, Length[l]-1}]][Total[l]];

f[l_] := Function[n, If[n<2 || l[[-1]]==1, 1, If[l[[1]]==0, 0, If[n==2, Binomial[l[[1]] + l[[2]], l[[1]]]-1, g[l]]]]][Length[l]];

a[n_] := f[Range[n]];

Table[a[n], {n, 0, 11}] (* Jean-Fran├žois Alcover, Feb 27 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A000217, A000330, A047909, A268485.

Sequence in context: A159318 A318132 A134795 * A273381 A094680 A259326

Adjacent sequences:  A268664 A268665 A268666 * A268668 A268669 A268670

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 10 2016

STATUS

approved

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Last modified September 18 20:55 EDT 2019. Contains 327181 sequences. (Running on oeis4.)