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A268698 Total number of sequences with p_j copies of j and longest increasing subsequence of length k summed over all partitions [p_1, p_2, ..., p_k] of n. 4
1, 1, 2, 4, 13, 36, 157, 554, 2800, 12530, 70772, 362188, 2370564, 13658713, 95366064, 642861687, 4774830263, 34769374156, 288999332899, 2255537559077, 19693313843687, 172690825379198, 1572921138465599, 14538979953843188, 145980379536597239 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..60

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

EXAMPLE

The partitions of 4 are [1,1,1,1], [2,1,1], [2,2], [3,1], [4] giving the a(4) = 13 sequences: 1234, 1123, 1213, 1231, 1122, 1212, 1221, 2112, 2121, 1112, 1121, 1211, 1111.

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)):

h:= (n, i, l)-> `if`(n=0 or i=1, f([1$n, l[]]), h(n, i-1, l)+

                `if`(i>n, 0, h(n-i, i, [i, l[]]))):

a:= n-> h(n$2, []):

seq(a(n), n=0..25);

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]]; h[n_, i_, l_] := h[n, i, l] = If[n == 0 || i == 1, f[Join[Array[1&, n], l]], h[n, i - 1, l] + If[i>n, 0, h[n-i, i, Prepend[l, i]]]]; a[n_] := h[n, n, {}]; Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, Feb 02 2017, translated from Maple *)

CROSSREFS

Cf. A047909, A268699, A268700, A268701.

Sequence in context: A148251 A144924 A148252 * A174604 A148253 A148254

Adjacent sequences:  A268695 A268696 A268697 * A268699 A268700 A268701

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Feb 11 2016

STATUS

approved

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Last modified September 29 00:44 EDT 2022. Contains 357081 sequences. (Running on oeis4.)