A366525 Irregular triangular array read by rows: T(n,k) = number of partitions p of n such that f(p) = k >= 0, where f is defined in Comments.

1, 1, 1, 2, 1, 2, 3, 4, 3, 3, 6, 2, 6, 6, 3, 5, 9, 5, 3, 7, 9, 9, 4, 1, 7, 13, 12, 6, 4, 10, 12, 15, 12, 5, 2, 7, 16, 19, 16, 12, 5, 2, 12, 16, 24, 22, 18, 6, 3, 11, 20, 28, 29, 24, 14, 6, 3, 12, 19, 31, 34, 36, 24, 13, 4, 3, 12, 23, 36, 42, 50, 30, 25, 8, 4

Offset: 1

Author

Clark Kimberling, Oct 12 2023

Keywords

• nonn
• tabf

Comments

For a partition p = (p(1),p(2),...,p(k)) of n, where p(1) >= p(2) >= ... >= p(k), define r(p) by subtracting 1 from p(1) and adding 1 to p(k) and then arranging the result in nonincreasing order. Iterating r eventually results in repetition; the function f(p) is the number of iterations of r up to but not including the first repeat. For example, starting with (5,3,2,2,1,1,1,1), the r-iterates are (4,3,2,2,2,1,1,1), (3,3,2,2,2,2,1,1), (3,2,2,2,2,2,2,1), (2,2,2,2,2,2,2,2), (3,2,2,2,2,2,2,1), so that f(5,3,2,2,1,1,1,1) = 3.

Examples

			First 18 rows:
   1
   1     1
   2     1
   2     3
   4     3
   3     6     2
   6     6     3
   5     9     5     3
   7     9     9     4     1
   7    13    12     6     4
  10    12    15    12     5     2
   7    16    19    16    12     5    2
  12    16    24    22    18     6    3
  11    20    28    29    24    14    6      3
  12    19    31    34    36    24    13     4     3
  12    23    36    42    50    30    25     8     4    1
  16    23    42    54    59    45    34    15     5    4
  13    28    47    57    74    61    52    28    16    5    4
Row 6 represents 3 partitions p that are self-repeating (i.e., k = 0), 6 such that f(p) = 1, and 2 such that f(p) = 2. Specifically,
  f(p) = 0 for these partitions: [6], [2,2,1,1], [2,1,1,1].
f(p) = 1 for these: [4,2], [3,3], [3,2,1], [3,1,1,1], [2,2,2], [1,1,1,1,1,1].
f(p) = 2 for these: [5,1], [4,1,1].
		

Links

• John Tyler Rascoe, Rows n = 1..60, flattened

Crossrefs

Cf. A000041 (row sums), A003479 (row lengths, after 2nd term).

Cf. A062968 (1st column).

Programs

Mathematica

r[list_] := If[Length[list] == 1, list, Reverse[Sort[# +
Join[{-1}, ConstantArray[0, Length[#] - 2], {1}]] &[Reverse[Sort[list]]]]];
f[list_] := NestWhileList[r, Reverse[Sort[list]], Unequal, All];
t = Table[BinCounts[#, {0, Max[#] + 1, 1}] &[Map[-1 + Length[Union[#]] &[f[#]] &, IntegerPartitions[n]]], {n, 1,20}]
Map[Length, t]; t1 = Take[t, 18]; TableForm[t1]
Flatten[t1]
(* Peter J. C. Moses, Oct 10 2023 *)

Python

from sympy .utilities.iterables import ordered_partitions
from collections import Counter
def A366525_rowlist(row_n):
    A = []
    for i in range(1,row_n+1):
        A.append([]); p,C = list(ordered_partitions(i)),Counter()
        for j in range(0,len(p)):
            x,a1,a,b = 0,[],list(p[j]),list(p[j])
            while i:
                b[-1] -= 1; b[0] += 1
                if b[-1] == 0: b.pop(-1)
                b = sorted(b); x += 1
                if a == b or a1 == b:
                    C.update({x}); break
                else:
                    a1 = a.copy(); a = b.copy()
        for z in range(1,len(C)+1): A[i-1].append(C[z])
    return(A) # John Tyler Rascoe, Nov 09 2023