A370368 -id:A370368 - cp's OEIS frontend

A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 9, 8, 0, 0, 1, 0, 34, 252, 60, 0, 0, 1, 0, 125, 5672, 14337, 544, 0, 0, 1, 0, 461, 125750, 2604732, 1327104, 6040, 0, 0, 1, 0, 1715, 2857472, 488360625, 2533087904, 182407545, 79008, 0, 0

Offset: 0

Alois P. Heinz, Feb 16 2024

			A(2,3) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
Square array A(n,k) begins:
  1, 1,   1,       1,          1,             1, ...
  0, 0,   0,       0,          0,             0, ...
  0, 0,   2,       9,         34,           125, ...
  0, 0,   8,     252,       5672,        125750, ...
  0, 0,  60,   14337,    2604732,     488360625, ...
  0, 0, 544, 1327104, 2533087904, 5192229797500, ...

Alois P. Heinz, Antidiagonals n = 0..54, flattened
Wikipedia, Partition of a set

Columns k=0+1,2-3 give: A000007, A053871, A370357.
Rows n=0-2 give: A000012, A000004, A010763(n-1) for k>0.
Main diagonal gives A370367.
Antidiagonal sums give A370368.
Cf. A060540, A370363.

A:= proc(n, k) `if`(k=0,`if`(n=0, 1, 0), add(
      (-1)^(n-j)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = A060540(n,k) - A370363(n,k) for n,k >= 1.

A370365 Total sum over all j in [n] of the number of partitions of [j*(n-j)] into (n-j) sets of size j having at least one set of consecutive numbers whose maximum (if j>0) is a multiple of j.

Original entry on oeis.org

0, 1, 2, 3, 4, 11, 77, 1571, 101924, 21824842, 18998281193, 63437859518312, 1037654210033812290, 72422876152852051595343, 27306605231809196751929593081, 50723306700937648229840111395656830, 510196838745355443955126736574361550469276

Offset: 0

Alois P. Heinz, Feb 16 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..55
Wikipedia, Partition of a set

Antidiagonal sums of A370363.

Cf. A370368, A370407.

b:= proc(n, k) option remember; `if`(k=0, signum(n), add(
      (-1)^(n-j+1)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n-1))
    end:
a:= n-> add(b(j, n-j), j=0..n):
seq(a(n), n=0..16);

a(n) = Sum_{j=0..n} A370363(j,n-j).

a(n) = A370407(n) - A370368(n).

A370407 Total sum over all j in [n] of the number of partitions of [j*(n-j)] into (n-j) sets of size j.

Original entry on oeis.org

1, 2, 3, 4, 7, 29, 424, 22250, 4166012, 3228619112, 9836415861419, 148021077093705105, 9516162824804128833773, 3369338041967340627557507931, 5792066385997100947453116161699033, 55416753515944143275546728017602371379095

Offset: 0

Alois P. Heinz, Feb 17 2024

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..54

Cf. A060540, A275973, A361948, A370365, A370368.

a:= n-> add(`if`(j=n, 1, (j*(n-j))!/((n-j)!^j*j!)), j=0..n):
seq(a(n), n=0..15);

a(n) = A370365(n) + A370368(n).
a(n) = Sum_{j=0..n} A361948(j,n-j).
a(n) mod 2 = A275973(n-1) for n>=2.

cp's OEIS Frontend

A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A370365 Total sum over all j in [n] of the number of partitions of [j*(n-j)] into (n-j) sets of size j having at least one set of consecutive numbers whose maximum (if j>0) is a multiple of j.

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Author

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Crossrefs

Programs

Maple

Formula

A370407 Total sum over all j in [n] of the number of partitions of [j*(n-j)] into (n-j) sets of size j.

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Author

Keywords

Links

Crossrefs

Programs

Maple

Formula