A361043 Array read by descending antidiagonals. A(n, k) is, if n > 0, the number of multiset permutations of {0, 1} of length n * k where the number of occurrences of 1 are multiples of n. A(0, k) = k + 1.

1, 2, 1, 3, 2, 1, 4, 4, 2, 1, 5, 8, 8, 2, 1, 6, 16, 32, 22, 2, 1, 7, 32, 128, 170, 72, 2, 1, 8, 64, 512, 1366, 992, 254, 2, 1, 9, 128, 2048, 10922, 16512, 6008, 926, 2, 1, 10, 256, 8192, 87382, 261632, 215766, 37130, 3434, 2, 1, 11, 512, 32768, 699050, 4196352, 6643782, 2973350, 232562, 12872, 2, 1

Offset: 0

Author

Peter Luschny, Mar 18 2023

Keywords

• nonn
• tabl

Comments

Because of the interchangeability of 0 and 1 in the definition, A(n, k) is even if n, k >= 1. In other words, if the binary representation of a permutation of the defined type is counted, then so is the 1's complement of that representation.

Examples

			Array A(n, k) starts:
 [0] 1, 2,    3,      4,        5,          6,            7, ...  A000027
 [1] 1, 2,    4,      8,       16,         32,           64, ...  A000079
 [2] 1, 2,    8,     32,      128,        512,         2048, ...  A081294
 [3] 1, 2,   22,    170,     1366,      10922,        87382, ...  A007613
 [4] 1, 2,   72,    992,    16512,     261632,      4196352, ...  A070775
 [5] 1, 2,  254,   6008,   215766,    6643782,    215492564, ...  A070782
 [6] 1, 2,  926,  37130,  2973350,  174174002,  11582386286, ...  A070967
 [7] 1, 2, 3434, 232562, 42484682, 4653367842, 644032289258, ...  A094211
.
Triangle T(n, k) starts:
 [0]  1;
 [1]  2,   1;
 [2]  3,   2,    1;
 [3]  4,   4,    2,     1;
 [4]  5,   8,    8,     2,      1;
 [5]  6,  16,   32,    22,      2,      1;
 [6]  7,  32,  128,   170,     72,      2,     1;
 [7]  8,  64,  512,  1366,    992,    254,     2,    1;
 [8]  9, 128, 2048, 10922,  16512,   6008,   926,    2, 1;
 [9] 10, 256, 8192, 87382, 261632, 215766, 37130, 3434, 2, 1;
.
A(2, 2) = 8 = card(0000, 1100, 1010, 1001, 0110, 0101, 0011, 1111).
A(1, 3) = 8 = card(000, 100, 010, 001, 110, 101, 011, 111).
		

Crossrefs

Rows: A000027 (n=0), A000079 (n=1), A081294 (n=2), A007613 (n=3), A070775 (n=4), A070782 (n=5), A070967 (n=6), A094211 (n=7), A070832 (n=8), A094213 (n=9), A070833 (n=10).

Variant: A308500 (upwards antidiagonals).

Cf. A167009 (main diagonal).

Programs

Maple

T := (n, k) -> add(binomial((n - k)*k, j*k), j = 0 .. n-k):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);

SageMath

# In Python use this import:
# from sympy.utilities.iterables import multiset_permutations
def A(n: int, k: int) -> int:
    if n == 0: return k + 1
    count = 0
    for a in range(0, n * k + 1, n):
        S = [i < a for i in range(n * k)]
        count += Permutations(S).cardinality()
    return count
def ARow(n: int, size: int) -> list[int]:
    return [A(n, k) for k in range(size)]
for n in range(6): print(ARow(n, 5))

Formula

A(n, k) = Sum_{j=0..k} binomial(n*k, n*j).

T(n, k) = Sum_{j=0..n-k} binomial((n - k)*k, j*k).