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

1,4

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.

Links

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

Example

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].

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 *)

Prog

(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

Crossrefs

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

Cf. A062968 (1st column).

Sequence in context: A169809 A214962 A350809 * A360737 A076258 A259576

Adjacent sequences: A366522 A366523 A366524 * A366526 A366527 A366528

Keyword

nonn,tabf

Author

Clark Kimberling, Oct 12 2023

Status

approved