A385814 -id:A385814 - cp's OEIS frontend

A268193 Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of partitions of n which have k distinct parts i such that i+1 is also a part.

1, 2, 2, 1, 4, 1, 4, 3, 8, 2, 1, 8, 6, 1, 13, 7, 2, 15, 11, 4, 22, 15, 4, 1, 24, 24, 7, 1, 37, 26, 12, 2, 40, 42, 16, 3, 57, 50, 22, 6, 64, 72, 33, 6, 1, 89, 84, 46, 11, 1, 98, 122, 60, 15, 2, 135, 141, 82, 24, 3, 149, 198, 106, 32, 5, 199, 231, 144, 45, 8, 224, 309, 187, 61, 10, 1

Offset: 1

Emeric Deutsch, Feb 13 2016

T(n,k) = number of partitions of n having k singleton parts other than the largest part. Example: T(5,1) = 3 because we have [4,1'], [3,2'], [2,2,1'] (the counted singletons are marked). These partitions are connected by conjugation to those in the definition.
From Gus Wiseman, Jul 10 2025: (Start)
Also the number of integer partitions of n with k maximal subsequences of consecutive parts not decreasing by 1 (anti-runs). For example, row n = 8 counts partitions with the following anti-runs:
  ((8))                ((3,3),(2))          ((3),(2,2),(1))
  ((4,4))              ((4),(3,1))          ((3),(2),(1,1,1))
  ((5,3))              ((5,2),(1))
  ((6,2))              ((4,2),(1,1))
  ((7,1))              ((2,2,2),(1,1))
  ((4,2,2))            ((2,2),(1,1,1,1))
  ((6,1,1))            ((2),(1,1,1,1,1,1))
  ((2,2,2,2))
  ((3,3,1,1))
  ((5,1,1,1))
  ((4,1,1,1,1))
  ((3,1,1,1,1,1))
  ((1,1,1,1,1,1,1,1))
(End)

			T(5,1) = 3 because we have [3,2], [2,2,1], and [2,1,1,1].
T(9,2) = 4 because we have [3,2',1,1,1,1'], [3,2,2',1,1'], [3,3,2',1'], and [4,3',2'] (the i's are marked).
Triangle starts:
  1;
  2;
  2,1;
  4,1;
  4,3;
  8,2,1;
  8,6,1;
From _Gus Wiseman_, Jul 11 2025: (Start)
Row n = 8 counts the following partitions by number of singleton parts other than the largest part:
  (8)                (5,3)        (4,3,1)
  (4,4)              (6,2)        (5,2,1)
  (4,2,2)            (7,1)
  (6,1,1)            (3,3,2)
  (2,2,2,2)          (3,2,2,1)
  (3,3,1,1)          (4,2,1,1)
  (5,1,1,1)          (3,2,1,1,1)
  (2,2,2,1,1)
  (4,1,1,1,1)
  (2,2,1,1,1,1)
  (3,1,1,1,1,1)
  (2,1,1,1,1,1,1)
  (1,1,1,1,1,1,1,1)
(End)

Alois P. Heinz, Rows n = 1..800, flattened

Row sums are A000041.
Row lengths are A003056.
For distinct parts instead of anti-runs we have A116608.
Column k = 1 is A116931.
For runs instead of anti-runs we have A384881.
The strict case is A384905.
The corresponding rank statistic is A356228, non-strict version A384906.
The proper case is A385814, runs A385815.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
Cf. A000009, A008284, A024786, A047993, A098859, A116674, A287170, A325325, A384882, A385213.

g := add(x^j*mul(1+t*x^i+x^(2*i)/(1-x^i), i = 1 .. j-1)/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 27)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n to 25 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i, t) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1, t or j>0)*
      `if`(t and j=1, x, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, false)):
seq(T(n), n=1..20);  # Alois P. Heinz, Feb 13 2016

b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, t || j > 0]*If[t && j == 1, x, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, False]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 21 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1!=#2+1&]]==k&]],{n,0,10},{k,0,n}] (* Delete zeros for A268193. Gus Wiseman, Jul 10 2025 *)

T(n,0) = A116931(n).
Sum_{k>=1} T(n, k) = A000041(n) (the partition numbers).
Sum_{k>=1} k*T(n,k) = A024786(n-1).
G.f.: G(t,x) = Sum_{j>=1} ((x^j/(1-x^j))*Product_{i=1..j-1} (1 + tx^i + x^{2i}/(1-x^i))).

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.

Original entry on oeis.org

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}]

cp's OEIS Frontend

A268193 Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of partitions of n which have k distinct parts i such that i+1 is also a part.

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