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

Author

Gus Wiseman, Jun 27 2020

Keywords

• nonn

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).

Examples

			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)
		

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.

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.

Programs

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}]

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

Extensions

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

More terms from Christian Sievers, May 08 2025