A384887 -id:A384887 - cp's OEIS frontend

A385815 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive elements decreasing by 0 or 1.

1, 0, 1, 0, 2, 0, 0, 3, 0, 0, 0, 4, 1, 0, 0, 0, 5, 2, 0, 0, 0, 0, 7, 4, 0, 0, 0, 0, 0, 8, 7, 0, 0, 0, 0, 0, 0, 10, 12, 0, 0, 0, 0, 0, 0, 0, 13, 16, 1, 0, 0, 0, 0, 0, 0, 0, 15, 25, 2, 0, 0, 0, 0, 0, 0, 0, 0, 18, 34, 4, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Jul 09 2025

			The partition (8,5,4,2,1) has maximal runs ((8),(5,4),(2,1)) so is counted under T(20,3).
The partition (8,5,3,2,2) has maximal runs ((8),(5),(3,2,2)) so is also counted under T(20,3).
Row n = 9 counts the following partitions:
  (9)                  (6,3)            (5,3,1)
  (5,4)                (7,2)
  (3,3,3)              (8,1)
  (4,3,2)              (4,4,1)
  (3,2,2,2)            (5,2,2)
  (3,3,2,1)            (6,2,1)
  (2,2,2,2,1)          (7,1,1)
  (3,2,2,1,1)          (4,2,2,1)
  (2,2,2,1,1,1)        (4,3,1,1)
  (3,2,1,1,1,1)        (5,2,1,1)
  (2,2,1,1,1,1,1)      (6,1,1,1)
  (2,1,1,1,1,1,1,1)    (3,3,1,1,1)
  (1,1,1,1,1,1,1,1,1)  (4,2,1,1,1)
                       (5,1,1,1,1)
                       (4,1,1,1,1,1)
                       (3,1,1,1,1,1,1)
Triangle begins:
   1
   0   1
   0   2   0
   0   3   0   0
   0   4   1   0   0
   0   5   2   0   0   0
   0   7   4   0   0   0   0
   0   8   7   0   0   0   0   0
   0  10  12   0   0   0   0   0   0
   0  13  16   1   0   0   0   0   0   0
   0  15  25   2   0   0   0   0   0   0   0
   0  18  34   4   0   0   0   0   0   0   0   0
   0  23  46   8   0   0   0   0   0   0   0   0   0
   0  26  62  13   0   0   0   0   0   0   0   0   0   0
   0  31  82  22   0   0   0   0   0   0   0   0   0   0   0

Row sums are A000041, strict A000009.
Column k = 1 is A034296 (flat or gapless partitions, ranks A066311 or A073491).
For subsets instead of partitions we have A034839, anti-runs A384893.
The strict case appears to be A116674.
For anti-runs instead of runs we have A268193.
The corresponding rank statistic is A287170.
For proper runs instead of runs we have A384881.
For proper anti-runs instead of runs we have A385814.
A007690 counts partitions with no singletons (ranks A001694), complement A183558.
A047993 counts partitions with max part = length, rank A106529.
A098859 counts Wilf partitions, complement A336866 (ranks A325992).
A116608 counts partitions by distinct parts.
A116931 counts sparse partitions, ranks A319630.
Cf. A001227, A008284, A325325, A356226, A384884, A384885, A384887, A384905.

Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1<=#2+1&]]==k&]],{n,0,20},{k,0,n}]

A385214 Number of subsets of {1..n} without all equal lengths of maximal runs of consecutive elements increasing by 1.

Original entry on oeis.org

0, 0, 0, 0, 2, 8, 25, 66, 159, 361, 791, 1688, 3539, 7328, 15040, 30669, 62246, 125896, 253975, 511357, 1028052

Offset: 0

Gus Wiseman, Jun 25 2025

			The maximal runs of S = {1,2,4,5,6,8,9} are ((1,2),(4,5,6),(8,9)), with lengths (2,3,2), so S is counted under a(9).
The a(0) = 0 through a(5) = 8 subsets:
  .  .  .  .  {1,2,4}  {1,2,4}
              {1,3,4}  {1,2,5}
                       {1,3,4}
                       {1,4,5}
                       {2,3,5}
                       {2,4,5}
                       {1,2,3,5}
                       {1,3,4,5}

These subsets are ranked by A164708, complement A164707
The complement is counted by A243815.
For distinct instead of equal lengths we have A384176, complement A384175.
For anti-runs instead of runs we have complement of A384889, for partitions A384888.
For permutations instead of subsets we have complement of A384892, distinct A384891.
For partitions instead of subsets we have complement of A384904, strict A384886.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A049988 counts partitions with equal run-lengths, distinct A325325.
A329738 counts compositions with equal run-lengths, distinct A329739.
A384177 counts subsets with all distinct lengths of maximal anti-runs, ranks A384879.
A384887 counts partitions with equal lengths of gapless runs, distinct A384884.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A010027, A044813, A164710, A383013, A384885.

Table[Length[Select[Subsets[Range[n]],!SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

cp's OEIS Frontend

A385815 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive elements decreasing by 0 or 1.

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