A136102 -id:A136102 - cp's OEIS frontend

A136624 Irregular triangle read by rows: classify each numeric partition by sum of its parts and by the size of the staircase Ferrers board required to contain it. The triangle gives the number of partitions in each class, cf. A136102 and A136103.

1, 1, 2, 1, 2, 3, 3, 1, 2, 2, 6, 7, 6, 4, 1, 2, 2, 4, 8, 12, 15, 17, 14, 10, 5, 1, 2, 2, 4, 6, 12, 15, 23, 30, 39, 42, 40, 35, 25, 15, 6, 1, 2, 2, 4, 6, 10, 16, 23, 29, 42, 56, 71, 88, 103, 112, 114, 102, 86, 65, 41, 21, 7, 1, 2, 2, 4, 6, 10, 14, 24, 31, 43

Offset: 0

Alford Arnold, Jan 17 2008

Sequences A136102 and A136103 encode the numeric partitions by least prime signature and the Ferrers boards by 1 2 12 360 75600 174636000 ... A006939.

			Starting a new row each time we are required to use a larger Ferrer board the triangle begins:
  1
  ..1
  .....2...1
  .........2...3...3...1
  .............2...2...6...7...6...4...1
  .................2...2...4...8..12..15..17..14..10...5...1
  .....................2...2...4...6..12..15..23..30..39..42..40..35..25..15..6..1

John Tyler Rascoe, Rows n = 0..16, flattened

Cf. A000041 (column sums), A000108, A006939, A025487, A071724 (row sums), A136102, A136103, A136625.

d(s,n) = {my(v = setminus([1..n],s), r=[], c=1); for(i=2, #v, if(v[i]==v[i-1]+1, c++ , r=concat(r, c); c=1)) ; return(concat(r, c))}
tri(n) = {n*(n+1)/2}
S(n) = {my(R = x^tri(n)); if(n<1, return(1), for(i=1,n-1, forsubset([n,i], s, my(u=d(vecextract([1..n],s),n)); R+=(x^(tri(n)-sum(j=1,#u, tri(u[j]))))*prod(j=1,#u, sum(z=0,u[j]-1, S(z))))); return(R))}
A136624(row_n) = {Vecrev(S(row_n)/x^(row_n))} \\ John Tyler Rascoe, Feb 25 2025

a(26) onwards from John Tyler Rascoe, Feb 25 2025

A136103 A Catalan shaped array (A000245) of least prime signatures (A025487) based on staircase Ferrers boards; cf. A136102.

Original entry on oeis.org

1, 2, 4, 6, 12, 8, 24, 30, 36, 60, 72, 120, 180, 360, 16, 48, 144, 210, 216, 240, 420, 432, 720, 840, 900, 1080, 1260, 1680, 1800, 2160, 2520, 3600, 5040, 5400, 6300, 7560, 10800, 12600, 15120, 25200, 37800, 75600

Offset: 0

Alford Arnold, Jan 12 2008

			The array begins
1
2
4...6..12
8..24..30..36..60..72..120..180..360
16...
with 1 1 3 9 28 90 ... least prime signatures per row;
hence there are 1 2 5 14 42 132 ... Catalan objects counted.
The staircases can be coded 1 2 12 360 75600 174636000 ...: A006939.
There are 14 signatures dividing 360; but 5 of these also divide 12 so there are nine new signatures: 8..24..30..36..60..72..120..180..360

Cf. A000108, A000245, A006939, A025487, A136102.

cp's OEIS Frontend

A136624 Irregular triangle read by rows: classify each numeric partition by sum of its parts and by the size of the staircase Ferrers board required to contain it. The triangle gives the number of partitions in each class, cf. A136102 and A136103.

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