A268751 -id:A268751 - cp's OEIS frontend

A269129 Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding the pattern 12...n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 5, 1, 0, 0, 1, 43, 23, 1, 0, 0, 1, 374, 1879, 119, 1, 0, 0, 1, 3199, 173891, 102011, 719, 1, 0, 0, 1, 26945, 16140983, 117392909, 7235651, 5039, 1, 0, 0, 1, 224296, 1474050783, 142951955371, 117108036719, 674641325, 40319, 1

Offset: 0

Alois P. Heinz, Feb 19 2016

			Square array A(n,k) begins:
  0,   0,      0,         0,            0,               0, ...
  1,   0,      0,         0,            0,               0, ...
  1,   1,      1,         1,            1,               1, ...
  1,   5,     43,       374,         3199,           26945, ...
  1,  23,   1879,    173891,     16140983,      1474050783, ...
  1, 119, 102011, 117392909, 142951955371, 173996758190594, ...

Alois P. Heinz, Antidiagonals n = 0..50, flattened
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Columns k=0-10 give: A057427, A033312, A267532, A269113, A269114, A269115, A269116, A269117, A269118, A269119, A269120.
Rows n=0-10 give: A000004, A000007, A000012, A269121, A269122, A269123, A269124, A269125, A269126, A269127, A269128.
Main diagonal gives: A268751.
Cf. A047909, A089759, A187783, A331562.

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=0, 1, `if`(l[1]=0, 0, `if`(n=1 or l[-1]=1, 1,
    `if`(n=2, binomial(l[1]+l[2], l[1])-1, g(l))))))(nops(l)):
A:= (n, k)-> (k*n)!/k!^n - f([k$n]):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
b:= proc(k, p, j, l, t) option remember;
      `if`(k=0, (-1)^t/l!, `if`(p<0, 0, add(b(k-i, p-1,
       j+1, l+i*j, irem(t+i*j, 2))/(i!*p!^i), i=0..k)))
    end:
A:= (n, k)-> (n*k)!*(1/k!^n-b(n, k-1, 1, 0, irem(n, 2))*n!):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Mar 03 2016

b[k_, p_, j_, l_, t_] := b[k, p, j, l, t] = If[k == 0, (-1)^t/l!, If[p < 0, 0, Sum[b[k-i, p-1, j+1, l + i j, Mod[t + i j, 2]]/(i! p!^i), {i, 0, k}]] ];
A[n_, k_] := (n k)! (1/k!^n - b[n, k-1, 1, 0, Mod[n, 2]] n!); Table[ Table[ A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 07 2016, after Alois P. Heinz *)

A(n,k) = A089759(k,n) - A047909(k,n) = A187783(n,k) - A047909(k,n).

A268485 Number of sequences with n copies each of 1,2,...,n and longest increasing subsequence of length n.

Original entry on oeis.org

1, 1, 5, 1306, 46922017, 449363984934526, 1878320344216429026862153, 5078529731893937404909347067888886466, 12324197596430667064913735085330208112438377122058241, 35544813569338447788721757701614208334438136486811525386710064098254294

Offset: 0

Alois P. Heinz, Feb 05 2016

nonn

			a(2) = 5: 1122, 1212, 1221, 2112, 2121.
a(3) = 1306: 111222333, 111223233, 111223323, ..., 332212113, 332212131, 332212311.

Alois P. Heinz, Table of n, a(n) for n = 0..27
J. D. Horton and A. Kurn, Counting sequences with complete increasing subsequences, Congressus Numerantium, 33 (1981), 75-80. MR 681905

Main diagonal of A047909.

Cf. A268667, A268751.

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([n$n]):
seq(a(n), n=0..8);
# second Maple program:
b:= proc(k, p, j, l, t) option remember;
      `if`(k=0, (-1)^t/l!, `if`(p<0, 0, add(b(k-i, p-1,
       j+1, l+i*j, irem(t+i*j, 2))/(i!*p!^i), i=0..k)))
    end:
a:= n-> n!*(n^2)!*b(n, n-1, 1, 0, irem(n, 2)):
seq(a(n), n=0..10);  # Alois P. Heinz, Mar 03 2016

b[k_, p_, j_, l_, t_] := b[k, p, j, l, t] = If[k==0, (-1)^t/l!, If[p<0, 0, Sum[b[k-i, p-1, j+1, l+i*j, Mod[t + i*j, 2]]/(i!*p!^i), {i, 0, k}]]];
a[n_] := n!*(n^2)!*b[n, n - 1, 1, 0, Mod[n, 2]];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Jun 18 2018, translated from 2nd Maple program *)

cp's OEIS Frontend

A269129 Number A(n,k) of sequences with k copies each of 1,2,...,n avoiding the pattern 12...n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula