A374985 -id:A374985 - cp's OEIS frontend

A374514 Number of n X n matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

1, 1, 3, 197, 732963, 289599115433, 19454710000290140631, 324252739440855086589750626125, 1839663535877691613435674541258128354870051, 4664717625821787781559533555514908690826684467996898799881, 6714190347498763079980307954946450922919624466513063316268554904936722083543

Offset: 0

Andrew Howroyd, Sep 16 2024

nonn

			The a(2) = 3 matrices are:
  [1 2]    [1 2]    [1 3]
  [2 3]    [3 4]    [2 4]

Ludovic Schwob, Table of n, a(n) for n = 0..20
Andrew Howroyd, PARI Program, Sep 2024.

Main diagonal of A374985.

Cf. A039622 (case all values also distinct), A068942, A376162 (case for symmetric matrices).

\\ See Links section for program file.
vector(8, n, A374514(n-1))

A375052 Number of n X 4 matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

Original entry on oeis.org

1, 1, 45, 4593, 732963, 155242003, 40007492715, 11910942902211, 3961395353371797, 1438326441446892453, 560815045483180502313, 231969721919492199940197, 100843924627856371008805767, 45742671500990337278105740455, 21524779658986553968372985081175

Offset: 0

Andrew Howroyd, Sep 16 2024

nonn

In other words, a(n) is the number of increasing tableaux of shape (n,n,n,n).

Andrew Howroyd, Table of n, a(n) for n = 0..40

Column k=4 of A374985.

Cf. A105124.

A378173 Array read by antidiagonals: T(n,k) is the number of proper antichain partitions of the rectangular poset of size n X k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 38, 14, 1, 1, 42, 372, 372, 42, 1, 1, 132, 4282, 14606, 4282, 132, 1, 1, 429, 55149

Offset: 1

Ludovic Schwob, Nov 18 2024

A partition of a poset into antichains is said to be proper if it does not contain two antichains A_1 and A_2, with x_1,y_1 in A_1 and x_2,y_2 in A_2, such that x_1y_2.

A proper antichain partition of a poset is endowed with an order relation, which is induced by the order relation of the poset. Let Y be a young diagram, and P the poset of shape Y. The number of linear extensions of P is the number of standard Young tableaux with shape Y. The sum over all proper antichain partitions of P, of the numbers of linear extensions of the induced orders, is equal to the number of normal generalized Young tableaux of shape Y with all rows and columns strictly increasing (cf. A299968).

			Array begins:
=====================================================================
n/k | 1     2      3      4      5      6 ...
----+----------------------------------------------------------------
  1 | 1     1      1      1      1      1 ...
  2 | 1     2      5     14     42    132 ...
  3 | 1     5     38    372   4282  55149 ...
  4 | 1    14    372  14606 ...
  5 | 1    42   4282 ...
  6 | 1   132  55149 ...

Cf. A060854, A299968, A374985.

T(n,k) = T(k,n).

T(n,2) = A000108(n).

cp's OEIS Frontend

A374514 Number of n X n matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

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A375052 Number of n X 4 matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

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A378173 Array read by antidiagonals: T(n,k) is the number of proper antichain partitions of the rectangular poset of size n X k.

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