A359413 Triangle read by rows: T(n, k) is the number of permutations of size n that require exactly k iterations of the pop-stack sorting map to reach the identity, for n >= 1, 0 <= k <= n-1.

1, 1, 1, 1, 3, 2, 1, 7, 8, 8, 1, 15, 26, 46, 32, 1, 31, 80, 191, 262, 155, 1, 63, 234, 735, 1440, 1737, 830, 1, 127, 664, 2752, 6924, 12314, 12432, 5106, 1, 255, 1850, 10114, 31928, 73122, 112108, 98156, 35346, 1, 511, 5088, 36564, 145199, 404758, 816401, 1104042, 844038, 272198

Offset

1,5

Comments

When k is fixed, T(n, k) has a rational g.f. (see A. Claesson and B. A. Guðmundsson).

Links

Bjarki Ágúst Guðmundsson, Rows n=1..16 of triangle, flattened

M. Albert and V. Vatter, How many pop-stacks does it take to sort a permutation?, arXiv:2012.05275 [math.CO], 2020.

A. Claesson and B. A. Guðmundsson, Enumerating permutations sortable by k passes through a pop-stack, arXiv:1710.04978 [math.CO], 2017-2019.

L. Pudwell and R. Smith, Two-stack-sorting with pop stacks, arXiv:1801.05005 [math.CO], 2018.

Peter Ungar, 2N noncollinear points determine at least 2N directions, J. Combin. Theory Ser. A, 33:3 (1982), pp. 343-347.

Formula

T(n, 0) = 1.

T(n, 1) = 2^(n-1)-1 for n >= 2 (see L. Pudwell and R. Smith).

T(n, 2) = A224232(n) - A011782(n) for n >= 3.

T(n, 3) = A293774(n) - A224232(n) for n >= 4.

T(n, 4) = A293775(n) - A293774(n) for n >= 5.

T(n, 5) = A293776(n) - A293775(n) for n >= 6.

T(n, 6) = A293784(n) - A293776(n) for n >= 7.

T(n, n-1) = A348905(n).

T(n, k) = 0 when k >= n (see M. Albert and V. Vatter).

Example

The pop-stack sorting map acts by reversing the descending runs of a permutation. For example, it sends 3412 to 3142, it sends 3142 to 1324, and it sends 1324 to 1234. This shows that if we start with the permutation 3412, then we require 4-1=3 iterations to reach the identity permutation. There are T(4,3) = 8 permutations of size 4 that require 3 iterations, namely 2341, 3241, 3412, 3421, 4123, 4132, 4231, 4312.
Triangle T(n,k) begins:
[1]  1;
[2]  1,  1;
[3]  1,  3,  2;
[4]  1,  7,  8,   8;
[5]  1, 15, 26,  46,  32;
[6]  1, 31, 80, 191, 262, 155;
...

Prog

(Python)
from itertools import permutations
def ps(lst):  # pop-stack sorting operator [cf. Claesson, Guðmundsson]
    out, stack = [], []
    for i in range(len(lst)):
        if len(stack) == 0 or stack[-1] < lst[i]:
            out.extend(stack[::-1])
            stack = []
        stack.append(lst[i])
    return out + stack[::-1]
def psops(t):
    c, lst, srtdlst = 0, list(t), sorted(t)
    if lst == srtdlst: return 0
    while lst != srtdlst:
        lst = ps(lst)
        c += 1
    return c
def T(n, k):
    return sum(1 for p in permutations(range(n), n) if psops(p) == k)
print([T(n, k) for n in range(1, 9) for k in range(n)]) # Michael S. Branicky, Nov 09 2021 (adapted from A348905 by Bjarki Ágúst Guðmundsson, Dec 30 2022)

Crossrefs

Cf. A011782, A224232, A224232, A293774, A293775, A293776, A293784, A348905.

Sequence in context: A130462 A059380 A145035 * A192020 A367564 A171128

Adjacent sequences: A359410 A359411 A359412 * A359414 A359415 A359416

Keyword

nonn,tabl

Author

Bjarki Ágúst Guðmundsson, Dec 30 2022

Status

approved