A000093 -id:A000093 - cp's OEIS frontend

A259449 Triangle read by rows arising from enumeration of partitions into non-integral powers.

1, 2, 5, 1, 8, 5, 11, 12, 1, 14, 23, 5, 18, 39, 16, 22, 62, 38, 3, 27, 91, 74, 12, 31, 127, 133, 33

Offset: 1

N. J. A. Sloane, Jun 27 2015

			Triangle begins:
1,
2,
5,1,
8,5,
11,12,1,
14,23,5,
18,39,16,
22,62,38,3,
27,91,74,12,
31,127,133,33,
...

B. K. Agarwala, F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. See Table 3.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

Columns include A000093, A000327.

A259450 Triangle read by rows arising from enumeration of partitions into non-integral powers.

Original entry on oeis.org

1, 4, 9, 3, 16, 14, 25, 39, 6, 36, 91, 32, 49, 173, 109, 8, 64, 307, 288, 47, 81, 502, 654, 194, 5, 100, 779, 1337, 592, 45

Offset: 1

N. J. A. Sloane, Jun 27 2015

			Triangle begins:
1,
4,
9,3,
16,14,
25,39,6,
36,91,32,
49,173,109,8,
64,307,288,47,
81,502,654,194,5,
100,779,1337,592,45,
...

B. K. Agarwala, F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. See Table 4.
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]

Columns include A000263, A000397.

A341822 Length of the longest 2-increasing sequence of positive integer triples with entries <= n.

Original entry on oeis.org

1, 2, 4, 8, 10, 14, 17, 21, 27, 30, 35

Offset: 1

Marcel K. Goh, Feb 20 2021

A triple t=(a_1,a_2,a_3) is defined to be 2-less than a triple u=(b_1,b_2,b_3) if a_i < b_i for at least two coordinates i. A sequence t^(j) of triples is 2-increasing if for all i < j, t^(i) is 2-less than t^(j).

Terms n <= 5 have been confirmed by brute-force search (Table 1 of Gowers and Long (2021)).

			For n=4, the sequence (1,1,1), (1,2,2), (2,1,3), (2,2,4), (3,3,1), (3,4,2), (4,3,3), (4,4,4) has length a(4)=8 and every 2-increasing sequence of length 9 must contain a triple with some coordinate equal to 5.

W. T. Gowers and J. Long, The length of an s-increasing sequence of r-tuples, Combinatorics, Probability and Computing 30 (2021), 1-36.

W. T. Gowers and J. Long, The length of an s-increasing sequence of r-tuples, arXiv:1609.08688 [math.CO], 2016.
Po-Shen Loh, Directed paths: from Ramsey to Ruzsa and Szemerédi, arXiv:1505.07312 [math.CO], 2015.

Cf. A000093.

a(n) >= n^{3/2} when n is a perfect square.

It is conjectured that a(n) <= n^{3/2} for all n.

Edited by N. J. A. Sloane, Mar 21 2021

a(10)-a(11) and confirmation of previous terms by Bert Dobbelaere, Mar 27 2021

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A259449 Triangle read by rows arising from enumeration of partitions into non-integral powers.

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A259450 Triangle read by rows arising from enumeration of partitions into non-integral powers.

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A341822 Length of the longest 2-increasing sequence of positive integer triples with entries <= n.

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