A370363 -id:A370363 - 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.

A370358 Number of partitions of [3n] into n sets of size 3 having at least one set {3j-2,3j-1,3j} (1<=j<=n).

Original entry on oeis.org

0, 1, 1, 28, 1063, 74296, 8182855, 1305232804, 284438292607, 81167321350432, 29367491879327959, 13135455977606994340, 7116140280642196449151, 4591529352468711908776288, 3479040085783649820897765223, 3058744793640846605215609362436

Offset: 0

Alois P. Heinz, Feb 16 2024

nonn

			a(1) = 1: 123.
a(2) = 1: 123|456.
a(3) = 28: 123|456|789, 123|457|689, 123|458|679, 123|459|678, 123|467|589, 123|468|579, 123|469|578, 123|478|569, 123|479|568, 123|489|567, 124|356|789, 125|346|789, 126|345|789, 127|389|456, 128|379|456, 129|378|456, 134|256|789, 135|246|789, 136|245|789, 137|289|456, 138|279|456, 139|278|456, 145|236|789, 146|235|789, 156|234|789, 178|239|456, 179|238|456, 189|237|456.

Alois P. Heinz, Table of n, a(n) for n = 0..223
Wikipedia, Partition of a set

Cf. A025035, A057427, A059841, A370347, A370357.

Column k=3 of A370363.

b:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
      9*(n*(n-1)/2*b(n-1)+(n-1)^2*b(n-2)+(n-1)*(n-2)/2*b(n-3)))
    end:
a:= n-> (3*n)!/(n!*(3!)^n)-b(n):
seq(a(n), n=0..20);

a(n) = Sum_{j=0..n-1} (-1)^(n-j+1) * binomial(n,j) * A025035(j).
a(n) = A025035(n) - A370357(n).
a(n) = Sum_{k=1..n} A370347(n,k).
a(n) mod 2 = A059841(n) for n>=2.
a(n) mod 9 = A057427(n).

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).

A370364 Number of partitions of [n^2] into n sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

Original entry on oeis.org

0, 1, 1, 28, 22893, 2443061876, 68542265471953355, 833412961429901104030214430, 6514551431426932053792271970458170132097, 45458343253887079540702419310885199704811913950207054152, 375236832464739513549091449370258959406125572044428827214970469920572831639

Offset: 0

Alois P. Heinz, Feb 16 2024

nonn

			a(1) = 1: 1.
a(2) = 1: 12|34.
a(3) = 28: 123|456|789, 123|457|689, 123|458|679, 123|459|678, 123|467|589, 123|468|579, 123|469|578, 123|478|569, 123|479|568, 123|489|567, 124|356|789, 125|346|789, 126|345|789, 127|389|456, 128|379|456, 129|378|456, 134|256|789, 135|246|789, 136|245|789, 137|289|456, 138|279|456, 139|278|456, 145|236|789, 146|235|789, 156|234|789, 178|239|456, 179|238|456, 189|237|456.

Alois P. Heinz, Table of n, a(n) for n = 0..27
Wikipedia, Partition of a set

Main diagonal of A370363.

Cf. A057599, A370367.

a:= n-> add((-1)^(n-j+1)*binomial(n, j)*(n*j)!/(j!*n!^j), j=0..n-1):
seq(a(n), n=0..10);

a(n) = Sum_{j=0..n-1} (-1)^(n-j+1)*binomial(n,j)*(n*j)!/(j!*n!^j).
a(n) = A370363(n,n).
a(n) = A057599(n) - A370367(n).

A370487 Number of partitions of [3n] into 3 sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

Original entry on oeis.org

1, 1, 7, 28, 103, 376, 1384, 5146, 19303, 72928, 277132, 1058146, 4056232, 15600898, 60174898, 232676278, 901620583, 3500409328, 13612702948, 53017895698, 206769793228, 807386811658, 3156148445578, 12350146091398, 48371405524648, 189615909656626, 743877799422154

Offset: 0

Alois P. Heinz, Feb 19 2024

nonn

			a(0) = 1: {}|{}|{}.
a(1) = 1: 1|2|3.
a(2) = 7: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 14|23|56, 15|26|34, 16|25|34.

Alois P. Heinz, Table of n, a(n) for n = 0..1663
Wikipedia, Partition of a set

Row n=3 of A370363.

Cf. A000079, A001700, A003409, A029651, A131577, A255738.

a:= n-> `if`(n=0, 1, 3*binomial(2*n-1,n)-2):
seq(a(n), n=0..27);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, 3*n*(n-1)+1, ((15*n^2
      -31*n+12)*a(n-1)-(3*n-2)*(4*n-6)*a(n-2))/((3*n-5)*n))
    end:
seq(a(n), n=0..27);

a(n) = 3*binomial(2*n-1,n) - 2 for n >= 1, a(0) = 1.
G.f.: ((3*x+1)*(4*x-1)+3*(1-x)*sqrt(1-4*x))/(2*(1-x)*(1-4*x)).
a(n) = A003409(n) - 2 = A029651(n) - 2 = 3*A001700(n-1) - 2 for n >= 1.
a(n) mod 2 = A255738(n+1).
a(n) mod 2 = 1 <=> n in { A131577 }.

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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A370358 Number of partitions of [3n] into n sets of size 3 having at least one set {3j-2,3j-1,3j} (1<=j<=n).

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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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A370364 Number of partitions of [n^2] into n sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

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A370487 Number of partitions of [3n] into 3 sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

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Programs

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