A356262 Partition triangle read by rows counting the irreducible permutations sorted by the partition type of their Lehmer code.

1, 1, 0, 1, 0, 2, 1, 0, 2, 1, 9, 1, 0, 2, 3, 24, 17, 24, 1, 0, 2, 3, 3, 98, 29, 23, 156, 91, 55, 1, 0, 2, 8, 4, 181, 43, 157, 113, 1085, 243, 418, 714, 360, 118, 1, 0, 2, 7, 11, 4, 300, 61, 317, 461, 398, 2985, 536, 1822, 4366, 417, 7684, 1522, 3904, 2788, 1262, 245, 1

Offset: 0

Author

Peter Luschny, Aug 01 2022

Keywords

• nonn
• tabf

Comments

This is the Eulerian statistics of permutations as defined in A355777 restricted to the irreducible permutations. This is a refinement of A356263, which can be seen as Euler's triangle restricted to irreducible permutations.

The ordering of the partitions is defined in A080577. See the comments in A356116 for the definition of the terms 'partition triangle' and 'reduced partition triangle'.

Examples

			[0] 1;
[1] 1;
[2] 0, 1;
[3] 0, 2, 1;
[4] 0, [2, 1], 9, 1;
[5] 0, [2, 3], [24, 17], 24, 1;
[6] 0, [2, 3, 3], [98,  29, 23], [156, 91], 55, 1;
[7] 0, [2, 8, 4], [181, 43, 157, 113], [1085, 243, 418], [714, 360], 118, 1;
Summing the bracketed terms reduces the triangle to A356263 .
.
The Lehmer mapping of the irreducible permutations to the partitions, case n = 4, k = 1: 2341 and 4123 map to the partition [3, 1], and 3412 map to the partition [2, 2]. Thus A356263(4, 1) = 2 + 1 = 3. Compare with the example in A355777.
.
The partition mapping of row 4:
[4] => 0
[3, 1] => 2
[2, 2] => 1
[2, 1, 1] => 9
[1, 1, 1, 1] => 1
		

Links

• Peter Luschny, Permutations with Lehmer, a SageMath Jupyter Notebook.

Crossrefs

Cf. A356263 (reduced triangle), A003319 (row sums).

Cf. A355777.

Programs

SageMath

import collections
def reducible(p) -> bool:
    return any(i for i in range(1, p.size())
        if all(p(j) < p(k)
            for j in range(1, i + 1)
                for k in range(i + 1, p.size() + 1)
    )   )
def perm_irreducible_stats(n: int):
    res = collections.defaultdict(int)
    for p in Permutations(n):
        if reducible(p): continue
        l = p.to_lehmer_code()
        c = [l.count(i) for i in range(len(p)) if i in l]
        res[Partition(reversed(sorted(c)))] += 1
    return sorted(res.items(), key=lambda x: len(x[0]))
@cached_function
def A356262_row(n):
    if n <= 1: return [1]
    return [0] + [v[1] for v in perm_irreducible_stats(n)]
def A356262(n, k): return A356262_row(n)[k]
for n in range(0, 8): print(A356262_row(n))