A202756 -id:A202756 - cp's OEIS frontend

A202751 Number of n X n nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

1, 2, 8, 62, 924, 26394, 1442764, 150786848, 30114993376, 11489639088218, 8372083277093216, 11649087077771471438, 30947648445392475219812, 156963868041535457457609234, 1519762800266538697863301357568

Offset: 1

R. H. Hardin, Dec 23 2011

nonn

			Some solutions for n=5:
  0 0 0 0 0    0 0 0 0 0    0 0 0 0 0    0 0 0 0 0
  0 0 0 1 1    0 0 0 1 1    0 0 0 1 1    0 0 0 0 0
  0 0 0 1 1    0 0 1 1 2    0 0 1 1 1    0 0 0 1 1
  0 1 1 1 1    0 0 1 2 3    0 1 1 2 2    0 0 1 2 2
  0 1 2 2 2    0 1 2 3 4    0 1 2 2 3    0 1 1 2 3

R. H. Hardin, Table of n, a(n) for n = 1..18
Dylan Heuer, Partial Alternating Sign Matrix Bijections and Dynamics, arXiv:2403.02242 [math.CO], 2024. See p. 6.
Dylan Heuer and Jessica Striker, Partial permutation and alternating sign matrix polytopes, arXiv:2012.09901 [math.CO], 2020.

Diagonal of A202756.

A202752 Number of n X 4 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

Original entry on oeis.org

1, 4, 17, 62, 184, 462, 1022, 2052, 3819, 6688, 11143, 17810, 27482, 41146, 60012, 85544, 119493, 163932, 221293, 294406, 386540, 501446, 643402, 817260, 1028495, 1283256, 1588419, 1951642, 2381422, 2887154, 3479192, 4168912, 4968777, 5892404

Offset: 1

R. H. Hardin, Dec 23 2011

nonn

Column 4 of A202756.

			Some solutions for n=5:
..0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0....0..0..0..0
..0..0..0..1....0..0..0..1....0..0..0..0....0..0..1..1....0..0..0..0
..0..0..0..1....0..0..1..1....0..0..0..1....0..0..1..1....0..0..0..1
..0..1..1..1....0..0..1..2....0..0..0..1....0..1..2..2....0..0..1..2
..0..1..1..2....0..1..2..3....0..1..1..2....0..1..2..2....0..1..2..2

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A202756.

Empirical: a(n) = (1/360)*n^6 + (1/30)*n^5 + (5/72)*n^4 - (1/6)*n^3 + (77/180)*n^2 + (19/30)*n.
Conjectures from Colin Barker, Jun 01 2018: (Start)
G.f.: x*(1 - 3*x + 10*x^2 - 8*x^3 + 2*x^4) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A202753 Number of n X 5 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

Original entry on oeis.org

1, 5, 31, 184, 924, 3809, 13197, 39675, 106357, 259669, 586829, 1242946, 2491516, 4763097, 8738111, 15461061, 26494977, 44126629, 71634979, 113637500, 176531380, 269049265, 402952089, 593884703, 862423461, 1235348661, 1747178785

Offset: 1

R. H. Hardin, Dec 23 2011

nonn

Column 5 of A202756.

			Some solutions for n=5:
..0..0..0..0..0....0..0..0..0..0....0..0..0..0..0....0..0..0..0..0
..0..1..1..1..1....0..0..0..0..0....0..0..1..1..1....0..0..0..0..1
..0..1..1..1..2....0..0..0..1..1....0..1..1..1..1....0..0..0..0..1
..0..1..1..2..3....0..0..0..1..2....0..1..1..2..2....0..1..1..1..1
..0..1..2..3..4....0..0..1..1..2....0..1..2..2..2....0..1..1..2..2

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A202756.

Empirical: a(n) = (1/302400)*n^10 + (1/10080)*n^9 + (1/1008)*n^8 + (1/336)*n^7 - (67/14400)*n^6 + (7/480)*n^5 + (1021/6048)*n^4 - (145/1008)*n^3 + (10519/12600)*n^2 - (367/420)*n + 1.
Conjectures from Colin Barker, Jun 01 2018: (Start)
G.f.: x*(1 - 6*x + 31*x^2 - 47*x^3 + 110*x^4 - 162*x^5 + 140*x^6 - 79*x^7 + 31*x^8 - 8*x^9 + x^10) / (1 - x)^11.
a(n) = 11*a(n-1) - 55*a(n-2) + 165*a(n-3) - 330*a(n-4) + 462*a(n-5) - 462*a(n-6) + 330*a(n-7) - 165*a(n-8) + 55*a(n-9) - 11*a(n-10) + a(n-11) for n>11.
(End)

A202754 Number of n X 6 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

Original entry on oeis.org

1, 6, 51, 462, 3809, 26394, 150777, 721382, 2964632, 10720688, 34811491, 103179440, 282848319, 724794396, 1751160378, 4017593748, 8804203831, 18519925138, 37551252015, 73653037370, 140173677721, 259538952486, 468599962315

Offset: 1

R. H. Hardin, Dec 23 2011

nonn

Column 6 of A202756.

			Some solutions for n=5:
  0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0
  0 0 1 1 1 1   0 0 0 1 1 1   0 0 0 0 0 0   0 0 0 1 1 1
  0 0 1 1 2 2   0 0 0 1 2 2   0 0 0 0 1 1   0 0 0 1 1 2
  0 0 1 1 2 2   0 0 0 1 2 3   0 0 1 1 1 2   0 0 1 2 2 3
  0 1 2 2 2 2   0 0 1 2 3 3   0 1 2 2 2 3   0 1 2 3 3 4

R. H. Hardin, Table of n, a(n) for n = 1..210
Robert Israel, Maple-assisted proof of empirical formula

seq((1/4572288000)*n^15 + (1/76204800)*n^14 + (41/130636800)*n^13 + (1/272160)*n^12 + (12631/653184000)*n^11 + (113/5443200)*n^10 + (2941/914457600)*n^9 + (661/381024)*n^8 + (1820467/326592000)*n^7 - (38281/10886400)*n^6 + (995867/16329600)*n^5 + (4181/68040)*n^4 - (253877/2646000)*n^3 + (233011/529200)*n^2 + (667/1260)*n, n=1..30); # Robert Israel, Jun 02 2019

Empirical: a(n) = (1/4572288000)*n^15 + (1/76204800)*n^14 + (41/130636800)*n^13 + (1/272160)*n^12 + (12631/653184000)*n^11 + (113/5443200)*n^10 + (2941/914457600)*n^9 + (661/381024)*n^8 + (1820467/326592000)*n^7 - (38281/10886400)*n^6 + (995867/16329600)*n^5 + (4181/68040)*n^4 - (253877/2646000)*n^3 + (233011/529200)*n^2 + (667/1260)*n.

Empirical formula verified (see link). - Robert Israel, Jun 02 2019

A202755 Number of nX7 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

Original entry on oeis.org

1, 7, 78, 1022, 13197, 150777, 1442764, 11408125, 75393424, 424992394, 2086954916, 9097630767, 35775808290, 128617512596, 427443971156, 1325376633243, 3864035126505, 10661515466643, 27995060608622, 70289431296342

Offset: 1

R. H. Hardin Dec 23 2011

nonn

Column 7 of A202756

			Some solutions for n=5
..0..0..0..0..0..0..0....0..0..0..0..0..0..0....0..0..0..0..0..0..0
..0..0..0..0..1..1..1....0..0..0..0..0..0..1....0..0..0..0..0..0..1
..0..1..1..1..1..2..2....0..0..0..0..1..1..2....0..0..0..1..1..1..2
..0..1..1..2..2..3..3....0..0..1..1..1..2..3....0..0..1..1..1..1..2
..0..1..1..2..2..3..4....0..1..1..2..2..3..3....0..0..1..2..2..2..2

R. H. Hardin, Table of n, a(n) for n = 1..210

Empirical: a(n) = (1/1520925880320000)*n^21 + (1/14485008384000)*n^20 + (17/5431878144000)*n^19 + (41/517321728000)*n^18 + (10847/9053130240000)*n^17 + (73/6897623040)*n^16 + (3646499/76046294016000)*n^15 + (340729/3621252096000)*n^14 + (186024683/217275125760000)*n^13 + (282751/26873856000)*n^12 + (25808261/724250419200)*n^11 + (2895163/86220288000)*n^10 + (493582091273/760462940160000)*n^9 + (2226946937/1034643456000)*n^8 - (263782703/2715939072000)*n^7 + (3455599541/129330432000)*n^6 - (31330211789/754427520000)*n^5 + (2559599737/7185024000)*n^4 - (3245092381/7334712000)*n^3 + (7079701/6350400)*n^2 - (28181/27720)*n + 1

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A202751 Number of n X n nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

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A202752 Number of n X 4 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

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A202753 Number of n X 5 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

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A202754 Number of n X 6 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

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A202755 Number of nX7 nonnegative integer arrays with each row and column increasing from zero by 0 or 1.

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