A089055 -id:A089055 - cp's OEIS frontend

A089054 Solution to the non-squashing boxes problem (version 1).

1, 2, 4, 8, 14, 23, 36, 54, 78, 109, 149, 199, 262, 339, 434, 548, 686, 849, 1043, 1269, 1535, 1842, 2199, 2607, 3078, 3613, 4225, 4915, 5700, 6581, 7576, 8686, 9934, 11321, 12871, 14585, 16493, 18596, 20925, 23481, 26303, 29392, 32788, 36492, 40553, 44972, 49799

Offset: 0

N. J. A. Sloane, Dec 04 2003

nonn

Given n boxes labeled 1..n, such that box i weighs i grams and can support a total weight of i grams; a(n) = number of stacks of boxes that can be formed such that no box is squashed.

Amanda Folsom, Youkow Homma, Jun Hwan Ryu, and Benjamin Tong, On a general class of non-squashing partitions, Discrete Mathematics 339 (2016) 1482-1506.
Oystein J. Rodseth, Sloane's box stacking problem, Discrete Math. 306 (2006), no. 16, 2005-2009.
Øystein J. Rødseth and James A. Sellers, Congruences modulo high powers of 2 for Sloane's box stacking function, Australasian Journal of Combinatorics, Volume 44 (2009), Pages 255-263.
N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, arXiv:math/0312418 [math.CO], 2003; Discrete Math., 294 (2005), 259-274.

Cf. A000123, A088567, A089055, A090631, A090632. Row sums of A090641.

max = 50; B[x_] = 1+x/(1-x) + Sum[x^(3 2^(k-1))/Product[(1-x^(2^j)), {j, 0, k}], {k, 1, Log[2, max]}] + O[x]^max;
A[x_] = (B[x]-x)/(1-x)^2;
CoefficientList[A[x], x] (* Jean-François Alcover, Sep 01 2018 *)

G.f.: (B(x)-x)/(1-x)^2, where B(x) = g.f. for A088567.

A089239 Triangle read by rows: T(n,k) (n >= 0, 0 <= k <= n) giving number of solutions to the n-box stacking problem in which exactly k boxes are used in the stack.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 9, 3, 0, 1, 6, 15, 17, 7, 0, 0, 1, 7, 21, 28, 14, 1, 0, 0, 1, 8, 28, 43, 25, 3, 0, 0, 0, 1, 9, 36, 62, 41, 7, 0, 0, 0, 0, 1, 10, 45, 86, 63, 13, 0, 0, 0, 0, 0, 1, 11, 55, 115, 93, 23, 0, 0, 0, 0, 0, 0, 1, 12, 66, 150, 132, 37, 0

Offset: 0

N. J. A. Sloane, Dec 11 2003

Given n+1 boxes labeled 0..n, such that box i weighs i grams and can support a total weight of i grams, T(n,k) = number of ways to form a stack of boxes such that no box is squashed.

			Triangle begins:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 9 3 0
1 6 15 17 7 0 0
1 7 21 28 14 1 0 0

N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Row sums give A089055. Columns give A000217, A005744, A089240.

cp's OEIS Frontend

A089054 Solution to the non-squashing boxes problem (version 1).

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