A335837 Number of normal patterns matched by integer partitions of n.

1, 2, 5, 9, 18, 31, 54, 89, 146, 228, 358, 545, 821, 1219, 1795, 2596, 3741, 5323, 7521, 10534, 14659, 20232, 27788, 37897, 51410, 69347, 93111, 124348, 165378, 218924, 288646, 379021, 495864, 646272, 839490, 1086693, 1402268, 1803786, 2313498, 2958530, 3773093

Offset

0,2

Comments

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

Links

Christian Sievers, Table of n, a(n) for n = 0..2000

Wikipedia, Permutation pattern

Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Example

The a(0) = 1 through a(4) = 18  pairs of a partition with a matched pattern:
  ()/()  (1)/()   (2)/()     (3)/()       (4)/()
         (1)/(1)  (2)/(1)    (3)/(1)      (4)/(1)
                  (11)/()    (21)/()      (31)/()
                  (11)/(1)   (21)/(1)     (31)/(1)
                  (11)/(11)  (21)/(21)    (31)/(21)
                             (111)/()     (22)/()
                             (111)/(1)    (22)/(1)
                             (111)/(11)   (22)/(11)
                             (111)/(111)  (211)/()
                                          (211)/(1)
                                          (211)/(11)
                                          (211)/(21)
                                          (211)/(211)
                                          (1111)/()
                                          (1111)/(1)
                                          (1111)/(11)
                                          (1111)/(111)
                                          (1111)/(1111)

Mathematica

mstype[q_]:=q/.Table[Union[q][[i]]->i, {i, Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@Subsets[y]]], {y, IntegerPartitions[n]}], {n, 0, 8}]

Prog

(PARI)
lista(n) = {
  my(v=vector(n+1, i, 1+x*O(x^n)));
  for(k=1, n,
    v=vector(n\(k+1)+1, i,
        (1-x^(i*k))/(1-x^k)*v[i] + sum(j=i, n\k, x^(j*k)*v[j+1]) +
        x^(k*i)/(1-x^k)^2*v[1] ) );
  Vec(v[1]) } \\ Christian Sievers, May 08 2025

Crossrefs

The version for compositions in standard order is A335454.

The version for compositions is A335456.

The version for Heinz numbers of partitions is A335549.

The contiguous case is A335838.

Patterns are counted by A000670 and ranked by A333217.

Patterns contiguously matched by prime indices are A335516.

Contiguous divisors are counted by A335519.

Minimal patterns avoided by prime indices are counted by A335550.

Cf. A000005, A056986, A108917, A269134, A333257, A334299, A335458, A335465.

Sequence in context: A107705 A278690 A335838 * A392002 A288578 A002883

Adjacent sequences: A335834 A335835 A335836 * A335838 A335839 A335840

Keyword

nonn

Author

Gus Wiseman, Jun 27 2020

Extensions

a(18) corrected by and a(19)-a(22) from Jinyuan Wang, Jun 27 2020

More terms from Christian Sievers, May 08 2025

Status

approved