A258152 -id:A258152 - cp's OEIS frontend

A258371 Triangle read by rows: T(n,k) is number of ways of arranging n indistinguishable points on an n X n square grid such that k rows contain at least one point.

1, 2, 4, 3, 54, 27, 4, 408, 1152, 256, 5, 2500, 22500, 25000, 3125, 6, 13830, 315900, 988200, 583200, 46656, 7, 72030, 3709545, 25882780, 40588905, 14823774, 823543, 8, 360304, 39024384, 535754240, 1766195200, 1657012224, 411041792, 16777216

Offset: 1

Adam J.T. Partridge, May 28 2015

Row sums give A014062, n >= 1.
Leading diagonal is A000312, n >= 1.
The triangle t(n,k) = T(n,k)/binomial(n,k) gives the number of ways to place n stones into the k X n grid of squares such that each of the k rows contains at least one stone. See A259051. One can use a partition array for this (and the T(n,k)) problem. See A258152. - Wolfdieter Lang, Jun 17 2015

			The number of ways of arranging eight pawns on a standard chessboard such that two rows contain at least one pawn is T(8,2)=360304.
Triangle T(n,k) begins:
n\k 1      2        3       4        5       6 ...
1:  1
2:  2      4
3:  3     54      27
4:  4    408    1152      256
5:  5   2500   22500    25000     3125
6:  6  13830  315900   988200   583200   46656
...
n = 7:  7  72030 3709545 25882780  40588905 14823774 823543,
n = 8:  8 360304 39024384 535754240 1766195200 1657012224 411041792 16777216.

Giovanni Resta, Table of n, a(n) for n = 1..1830 (first 60 rows)

Cf. A000312, A014062, A258152, A259051.

T[n_,k_]:= Binomial[n,k] * Sum[Multinomial@@ (Last/@ Tally[e]) * Times@@ Binomial[n,e], {e, IntegerPartitions[n, {k}]}]; Flatten@ Table[ T[n,k],{n,9}, {k,n}] (* Giovanni Resta, May 28 2015 *)

T(n,2) = binomial(n,2)*(binomial(2*n,n)-2). - Giovanni Resta, May 28 2015

A259051 Triangle T(n,m) for the number of ways to put n stones into an m X n square grid such that each of the m rows contains at least one stone.

Original entry on oeis.org

1, 1, 4, 1, 18, 27, 1, 68, 288, 256, 1, 250, 2250, 5000, 3125, 1, 922, 15795, 65880, 97200, 46656, 1, 3430, 105987, 739508, 1932805, 2117682, 823543, 1, 12868, 696864, 7653632, 31539200, 59179008, 51380224, 16777216, 1, 48618, 4540968, 75687696, 461828790, 1320099444, 1919564892, 1377495072, 387420489

Offset: 1

Wolfdieter Lang, Jun 18 2015

This is the triangle A258371(n, m)/binomial(n, m).

For the corresponding partition array see A258152.

			The triangle T(n, m) begins:
n\k 1     2       3        4         5         6           7
1:  1
2:  1     4
3:  1    18      27
4:  1    68     288      256
5:  1   250    2250     5000      3125
6:  1   922   15795    65880     97200      46656
7:  1  3430  105987   739508   1932805    2117682     823543
...
8:  1 12868  696864  7653632  31539200 59179008 51380224
16777216,
9:  1 48618 4540968 75687696 461828790 1320099444 1919564892 1377495072 387420489.
a(4, 2) = 68 from the sum 32 + 36 of the n=4 row of A258152 which belong to the partitions of 4 with m=2 parts, namely (1, 3) and (2, 2).

Giovanni Resta, Table of n, a(n) for n = 1..1830 (first 60 rows)

Cf. A258152, A258371.

T[n_, k_]:= Sum[Multinomial@@ (Last/@ Tally[e]) * Times@@ Binomial[n, e], {e, IntegerPartitions[n, {k}]}]; Flatten@ Table[ T[n, k], {n, 9}, {k, n}] (* Giovanni Resta, Jun 18 2015 *)

T(n, m) = sum over the A258152(n, k) entries corresponding to partitions of n with m parts; n >= 1, m = 1,2, ..., n.

T(n, m) = A258371(n, m)/binomial(n, m).

cp's OEIS Frontend

A258371 Triangle read by rows: T(n,k) is number of ways of arranging n indistinguishable points on an n X n square grid such that k rows contain at least one point.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula