A172645 -id:A172645 - cp's OEIS frontend

A334549 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 7, 1, 1, 1, 1, 19, 31, 19, 1, 1, 1, 1, 51, 175, 175, 51, 1, 1, 1, 1, 141, 991, 2371, 991, 141, 1, 1, 1, 1, 393, 5881, 32611, 32611, 5881, 393, 1, 1, 1, 1, 1107, 35617, 481381, 1084851, 481381, 35617, 1107, 1, 1

Offset: 0

Andrew Howroyd, May 09 2020

Equivalently, the number of n X k 0..2 arrays with row sums k and column sums n.

			Array begins:
====================================================================
n\k | 0 1   2     3       4          5            6            7
----|---------------------------------------------------------------
  0 | 1 1   1     1       1          1            1            1 ...
  1 | 1 1   1     1       1          1            1            1 ...
  2 | 1 1   3     7      19         51          141          393 ...
  3 | 1 1   7    31     175        991         5881        35617 ...
  4 | 1 1  19   175    2371      32611       481381      7343449 ...
  5 | 1 1  51   991   32611    1084851     39612501   1509893001 ...
  6 | 1 1 141  5881  481381   39612501   3680774301 360255871641 ...
  7 | 1 1 393 35617 7343449 1509893001 360255871641 ...
     ...
The T(3,2) = 7 matrices are:
  [0 0]  [ 0  0]  [ 0  0]  [ 1 -1]  [-1  1]  [ 1 -1]  [-1  1]
  [0 0]  [ 1 -1]  [-1  1]  [ 0  0]  [ 0  0]  [-1  1]  [ 1 -1]
  [0 0]  [-1  1]  [ 1 -1]  [-1  1]  [ 1 -1]  [ 0  0]  [ 0  0]

Andrew Howroyd, Table of n, a(n) for n = 0..350 (first 26 antidiagonals)

Columns k=0..14 are A000012, A000012, A002426, A172634, A172642, A172639, A172633, A172636, A172638, A172641, A172637, A172644, A172640, A172643, A172635.
Main diagonal is A172645.
Cf. A008300, A333901, A376935, A377063 (up to row permutations).

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

A050204 a(n) is the number of n X n matrices of 0's, 1's and -1's in which the entries in each row or column sum to 1.

Original entry on oeis.org

1, 2, 21, 1344, 628080, 2030271480, 49846153939785, 9350902053107858880, 13807842729396124460629536, 162526525004284727135887319788800, 15464071313054035722739424623204673044920, 12011244510939290610945342901003078567310643860160

Offset: 1

Eric W. Weisstein

nonn

Cf. A005130 (Robbins numbers).
Cf. A229165 (entries in each row or column sum to 1 and there are no adjacent -1's or 1's in any row or column).
Cf. A172645 (case row and column sums are zero).

Definition corrected by Murphy Waggoner and R. H. Hardin, Sep 24 2013
a(6)-a(8) from R. H. Hardin, Sep 25 2013
a(9)-a(12) from Andrew Howroyd, Feb 03 2021

A377064 Number of {-1,0,1} n X n matrices with all rows and columns summing to zero up to permutations of rows.

Original entry on oeis.org

1, 1, 2, 6, 117, 9691, 5328136, 18490880339, 437425741017623, 72201603445260460729, 85704032961379965273243136, 746789021667791689307771100458717, 48671404211097237572497575028382156068182, 24090982261278741928086824237920135192809826750606, 91765357087196227652413496510903409610512369965590600589002

Offset: 0

Andrew Howroyd, Oct 15 2024

nonn

Columns are not permutable.

Equivalently, the number of n X n 0..2 arrays with row sums n and column sums n up to permutation of rows.

Main diagonal of A377063.

Cf. A172645.

cp's OEIS Frontend

A334549 Array read by antidiagonals: T(n,k) is the number of {-1,0,1} n X k matrices with all rows and columns summing to zero.

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Formula

A050204 a(n) is the number of n X n matrices of 0's, 1's and -1's in which the entries in each row or column sum to 1.

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A377064 Number of {-1,0,1} n X n matrices with all rows and columns summing to zero up to permutations of rows.

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