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.
Original entry on oeis.org
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
Offset: 0
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.
-
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$2, []):
seq(a(n), n=0..25);
-
g[l_] := g[l] = Function[n, f[Most[l]]*Binomial[n-1, l[[-1]]-1] + Sum[f[ Sort[ ReplacePart[l, j -> l[[j]]-1]]], {j, 1, Length[l]-1}]][Total[l]]; f[l_] := Function[n, If[n<2 || l[[-1]] == 1, 1, If[l[[1]] == 0, 0, If[n == 2, Binomial[l[[1]] + l[[2]], l[[1]]]-1, g[l]]]]][Length[l]]; h[n_, i_, l_] := h[n, i, l] = If[n == 0 || i == 1, f[Join[Array[1&, n], l]], h[n, i - 1, l] + If[i>n, 0, h[n-i, i, Prepend[l, i]]]]; a[n_] := h[n, n, {}]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)
A268699
Total number of sequences with c_j copies of j and longest increasing subsequence of length k summed over all compositions [c_1, c_2, ..., c_k] of n.
Original entry on oeis.org
1, 1, 2, 6, 22, 95, 471, 2618, 16052, 107313, 775045, 6002106, 49536510, 433485429, 4004680967, 38912323570, 396393445096, 4221367056961, 46878865762185, 541660919690866, 6498811587848690, 80818650742133717, 1040037672241415947, 13829246515918840106
Offset: 0
The compositions of 4 are [1,1,1,1], [2,1,1], [1,2,1], [1,1,2], [2,2], [3,1], [1,3], [4] giving the a(4) = 22 sequences: 1234, 1123, 1213, 1231, 1223, 2123, 1232, 1233, 1323, 3123, 1122, 1212, 1221, 2112, 2121, 1112, 1121, 1211, 1222, 2122, 2212, 1111.
-
c:= l-> f(l)*nops(l)!/(v-> mul(coeff(v, x, j)!,
j=0..degree(v)))(add(x^i, i=l)):
g:= proc(l) option remember; (n-> f(l[1..nops(l)-1])*
binomial(n-1, l[-1]-1)+ add(f(sort(subsop(j=l[j]
-1, l))), j=1..nops(l)-1))(add(i, i=l))
end:
f:= l-> (n-> `if`(n<2 or l[-1]=1, 1, `if`(l[1]=0, 0, `if`(
n=2, binomial(l[1]+l[2], l[1])-1, g(l)))))(nops(l)):
h:= (n, i, l)-> `if`(n=0 or i=1, c([1$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);
-
c[l_] := f[l]*Length[l]!/Function[v, Product[Coefficient[v, x, j]!, {j, 0, Exponent[v, x]}]][Sum[x^i, {i, l}]];
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_] := If[n == 0 || i == 1, c[Join[Array[1&, n], l]], h[n, i-1, l] + If[i > n, 0, h[n-i, i, Join[{i}, l]]]];
a[n_] := h[n, n, {}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 06 2022, after Alois P. Heinz *)
A268700
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 into distinct parts.
Original entry on oeis.org
1, 1, 1, 3, 4, 14, 46, 111, 330, 1614, 7348, 21340, 98145, 379405, 2633085, 14871033, 57284558, 278927415, 1609313975, 8289565670, 74945364815, 522977754235, 2403799401259, 14180489136597, 83964652635668, 623008803758260, 3918144764978718, 46950727351392315
Offset: 0
The partitions of 4 into distinct parts are [3,1], [4] giving the a(4) = 4 sequences: 1112, 1121, 1211, 1111.
-
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>i*(i+1)/2, 0, `if`(n=0, f(l), h(n, i-1, l)
+`if`(i>n, 0, h(n-i, i-1, [i, l[]])))):
a:= n-> h(n$2, []):
seq(a(n), n=0..30);
-
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_] := If[n>i*(i+1)/2, 0, If[n==0, f[l], h[n, i-1, l] + If[i>n, 0, h[n-i, i-1, Join[{i}, l]]]]]; a[n_] := h[n, n, {}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 11 2017, translated from Maple *)
Showing 1-3 of 3 results.