A181196 -id:A181196 - cp's OEIS frontend

A181197 Number of 3 X n matrices containing a permutation of 1..3*n in increasing order rowwise, columnwise and (downwards) antidiagonally.

1, 1, 4, 29, 290, 3532, 49100, 750325, 12310294, 213446666, 3868253164, 72686739116, 1407643591804, 27964937748724, 567853691242796, 11751537336221989, 247263499985110046, 5279409371079693454, 114199628255736623996, 2499214354674134770354

Offset: 1

R. H. Hardin, Oct 10 2010

nonn

Row 3 of A181196.
Equivalently, the number of "truncated shifted standard Young tableaux" of shape ; in other words, if we shift the middle row to the right by one unit and the bottom row to the right by two units, we require that the resulting diagram be increasing as we read down or to the right.
To count these tableaux, observe that if we put the entry 2n + 2 + k in the last position of the second row, the bottom row must end with the entries 2n + 3 + k, ..., 3n. The remaining figure can be filled in arbitrarily; it is a shifted Young diagram of shape . Now apply the hook-length formula for shifted Young tableaux. (This argument is due to Greta Panova.)
a(n) is also the number of maximum packings of pattern
[5 6]
[3 4]
[1 2] in column-strict arrays of size 3 X n+1. - Ran Pan, Apr 13 2015
a(n) is also the number of standard Young tableaux of shape (n,n,n) (French notation) such that for any element T(i,j) in the tableau T, its upper element T(i+1,j) is larger than its right element T(i,j+1). - Ran Pan, Apr 13 2015

			All four 3 X 3 examples:
1..2..3....1..2..3....1..2..4....1..2..4
4..5..6....4..5..7....3..5..6....3..5..7
7..8..9....6..8..9....7..8..9....6..8..9

Alois P. Heinz, Table of n, a(n) for n = 1..220
Joerg Arndt, The a(4)=29 truncated shifted standard Young tableaux of shape [4,4,4].
J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012. - From _N. J. A. Sloane_, Oct 12 2012
Ran Pan, Problem 0, Project P.
Ping Sun, Enumeration of standard Young tableaux of shifted strips with constant width, arXiv:1506.07256 [math.CO], 24 Jun 2015.

Row n=3 of A227578. - Alois P. Heinz, Jul 17 2013

a:= n-> `if`(n<2, 1, add(((2*n+k-1)!*(n-k)*(n-k-1)) /
         (n!*(n-1)!*k!*(2*n-1)*(n+k)*(n+k-1)), k=0..n-2)):
seq(a(n), n=1..30);  # Alois P. Heinz, Jul 01 2012

Flatten[{1,Table[Sum[((2*n+k-1)!*(n-k)*(n-k-1))/(n!*(n-1)!*k!*(2*n-1)*(n+k)*(n+k-1)),{k,0,n-2}],{n,2,20}]}] (* Vaclav Kotesovec, Jul 21 2013 *)

a(n) = Sum_{k=0..n-2} ((2n+k-1)!*(n-k)*(n-k-1)) / (n!*(n-1)!*k!*(2n-1) * (n+k)*(n+k-1)) for n>=2, a(1) = 1.
Recurrence: (2*n-1)*(7*n-13)*n^2*a(n) = 2*(182*n^4 - 1185*n^3 + 2722*n^2 - 2625*n + 900)*a(n-1) + 3*(2*n-5)*(3*n-5)*(3*n-4)*(7*n-6)*a(n-2). - Vaclav Kotesovec, Jul 21 2013
a(n) ~ 3^(3*n+1/2)/(64*Pi*n^4). - Vaclav Kotesovec, Jul 21 2013

Formula and comments from Joel B. Lewis, Jul 25 2011

A227578 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component such that for each point (p_1,p_2,...,p_k) we have p_1<=p_2<=...<=p_k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 4, 1, 1, 1, 14, 29, 8, 1, 1, 1, 42, 290, 185, 16, 1, 1, 1, 132, 3532, 7680, 1257, 32, 1, 1, 1, 429, 49100, 456033, 238636, 8925, 64, 1, 1, 1, 1430, 750325, 34426812, 77767945, 8285506, 65445, 128, 1

Offset: 0

Alois P. Heinz, Jul 16 2013

Conjecture: column k is asymptotic to c(k) * (k+1)^(k*n)/n^((k^2-1)/2), where c(k) is a constant dependent only on k. - Vaclav Kotesovec, Jul 21 2013

			A(4,0) = 1: [()].
A(3,1) = 4: [(3),(0)], [(3),(1),(0)], [(3),(2),(0)], [(3),(2),(1),(0)].
A(2,2) = 5: [(2,2),(0,2),(0,0)], [(2,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(0,2),(0,0)], [(2,2),(1,2),(0,2),(0,1),(0,0)], [(2,2),(1,2),(1,1),(0,1),(0,0)].
A(1,3) = 1: [(1,1,1),(0,1,1),(0,0,1),(0,0,0)].
A(0,4) = 1: [(0,0,0,0)].
Square array A(n,k) begins:
  1,  1,    1,      1,        1,           1, ...
  1,  1,    1,      1,        1,           1, ...
  1,  2,    5,     14,       42,         132, ...
  1,  4,   29,    290,     3532,       49100, ...
  1,  8,  185,   7680,   456033,    34426812, ...
  1, 16, 1257, 238636, 77767945, 36470203156, ...

Alois P. Heinz, Antidiagonals n = 0..25, flattened
Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez and Olga Basova, Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions, Linear Algebra and its Applications, Volume 530, 1 October 2017, p. 414-444. This entry is mentioned in the Introduction, but it seems that actually they mean A181196.

Columns k=0-10 give: A000012, A011782, A059231, A227580, A227583, A227596, A227597, A227598, A227599, A227600, A227601.
Rows n=0+1, 2-10 give: A000012, A000108(k+1), A181197(k+2), A227584, A227602, A227603, A227604, A227605, A227606, A227607.
Main diagonal gives: A227579.
Cf. A060854 (steps decrement one component by 1), A262809, A263159.
A181196 is a similar but different array.

b:= proc(l) option remember; `if`(l[-1]=0, 1, add(add(b(subsop(
      i=j, l)), j=`if`(i=1, 0, l[i-1])..l[i]-1), i=1..nops(l)))
    end:
A:= (n, k)-> `if`(k=0, 1, b([n$k])):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[l_] := b[l] = If[ l[[-1]] == 0, 1, Sum[ Sum[ b[ReplacePart[l, i -> j]], {j, If[i == 1, 0, l[[i-1]]], l[[i]]-1}], {i, 1, Length[l]}]]; a[n_, k_] := If[k == 0, 1, b[Array[n&, k]]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 09 2013, translated from Maple *)

A181191 Number of n X n matrices containing a permutation of 1..n*n in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

Original entry on oeis.org

1, 1, 4, 169, 141696, 3777546912, 4673805856338368, 368253691037579094795185, 2426023001499238992505630883146240, 1697356437632520242875237327471631991584394752, 156101219875805260212264222801658705937606174957553142873088

Offset: 1

R. H. Hardin, Oct 10 2010

nonn

			All solutions for 3 X 3:
..1..2..3....1..2..3....1..2..4....1..2..4
..4..5..6....4..5..7....3..5..6....3..5..7
..7..8..9....6..8..9....7..8..9....6..8..9

Alois P. Heinz, Table of n, a(n) for n = 1..20

Main diagonal of A181196.

Cf. A039917, A181196.

b:= proc(l) option remember; local n; n:= nops(l);
      `if`({l[]}={0}, 1, add(`if`((i=1 or l[i-1]<=l[i]) and l[i]>
      `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l)), 0), i=1..n))
    end:
a:= n-> b([n$n]):
seq(a(n), n=1..10);  # Alois P. Heinz, Jul 24 2012

b[l_] := b[l] = With[{n = Length[l]},
   If[Union[l]=={0}, 1, Sum[If[(i==1 || l[[i-1]] <= l[[i]]) && l[[i]] >
   If[i==n, 0, l[[i+1]]], b[ReplacePart[l, i->l[[i]]-1]], 0], {i, 1, n}]]];
a[n_] := b[Table[n, {n}]];
Table[a[n], {n, 1, 11}] (* Jean-François Alcover, May 10 2022, after Alois P. Heinz *)

a(10)-a(16) from Alois P. Heinz, Jul 24 2012

A181192 Number of n X 5 matrices containing a permutation of 1..n*5 in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

Original entry on oeis.org

1, 14, 290, 6392, 141696, 3142704, 69705920, 1546100352, 34293030016, 760631058944, 16871055411200, 374205743270912, 8300010573582336, 184097055591849984, 4083335265314938880, 90569764059295875072

Offset: 1

R. H. Hardin, Oct 10 2010

nonn

Column 5 of A181196.

			Some solutions for 4 X 5:
..1..2..3..4..5....1..2..3..4..5....1..2..3..4..5....1..2..3..4..5
..6..7..8..9.10....6..7..8..9.10....6..7..8..9.10....6..7..8..9.10
.11.12.13.14.15...11.12.13.14.17...11.12.13.14.16...11.12.13.14.18
.16.17.18.19.20...15.16.18.19.20...15.17.18.19.20...15.16.17.19.20

R. H. Hardin, Table of n, a(n) for n=1..100

Cf. A181196.

Empirical: a(n) = 24*a(n-1) - 40*a(n-2) - 8*a(n-3).
Conjectures from Colin Barker, Feb 27 2018: (Start)
G.f.: x*(1 - 10*x - 6*x^2) / ((1 - 2*x)*(1 - 22*x - 4*x^2)).
a(n) = 2^(n-2) + ((11-5*sqrt(5))^n*(2+sqrt(5)) + (-2+sqrt(5))*(11+5*sqrt(5))^n) / (4*sqrt(5)).
(End)

A181193 Number of n X 6 matrices containing a permutation of 1..n*6 in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

Original entry on oeis.org

1, 42, 3532, 352184, 36372976, 3777546912, 392658046912, 40820345224064, 4243729567634176, 441184342397471232, 45866192670977108992, 4768319236090599225344, 495721734753595527294976

Offset: 1

R. H. Hardin, Oct 10 2010

nonn

Column 6 of A181196.

			Some solutions for 4 X 6:
..1..2..3..4..5..6....1..2..3..4..5..6....1..2..3..4..5..6....1..2..3..4..5..6
..7..8..9.10.11.12....7..8..9.10.11.12....7..8..9.10.11.12....7..8..9.10.11.12
.13.14.15.16.17.18...13.14.15.17.19.20...13.14.15.17.19.21...13.14.15.17.19.22
.19.20.21.22.23.24...16.18.21.22.23.24...16.18.20.22.23.24...16.18.20.21.23.24

R. H. Hardin, Table of n, a(n) for n=1..100

Cf. A181196.

Empirical: a(n) = 120*a(n-1) - 1672*a(n-2) + 544*a(n-3) - 6672*a(n-4) + 256*a(n-5).

Empirical g.f.: x*(1 - 78*x + 164*x^2 - 1976*x^3 + 224*x^4) / ((1 - 16*x)*(1 - 104*x + 4*x^2)*(1 + 4*x^2)). - Colin Barker, Feb 27 2018

A181194 Number of n X 7 matrices containing a permutation of 1..n*7 in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

Original entry on oeis.org

1, 132, 49100, 25097600, 14083834704, 8092149471168, 4673805856338368, 2702482348019033600, 1562998915785444034816, 904016986809953084011520, 522876616523380188491377664, 302428597238403007787949047808

Offset: 1

R. H. Hardin, Oct 10 2010

nonn

Column 7 of A181196.

			Some solutions for 4 X 7:
..1..2..3..4..5..6..7....1..2..3..4..5..7.10....1..2..3..4..5..7.10
..8..9.10.11.12.13.14....6..8..9.12.13.15.19....6..8..9.12.13.15.19
.15.16.17.18.19.20.21...11.14.17.18.20.22.24...11.14.17.18.20.22.25
.22.23.24.25.26.27.28...16.21.23.25.26.27.28...16.21.23.24.26.27.28

R. H. Hardin, Table of n, a(n) for n=1..100

Cf. A181196.

Empirical: a(n) = 720*a(n-1) - 84448*a(n-2) + 1503360*a(n-3) - 17912224*a(n-4) - 318223104*a(n-5) + 564996096*a(n-6) + 270471168*a(n-7) - 11373824*a(n-8) + 65536*a(n-9).

Empirical g.f.: x*(1 - 588*x + 38508*x^2 - 610624*x^3 - 20571792*x^4 + 98269760*x^5 + 112896832*x^6 - 6488576*x^7 + 57344*x^8) / ((1 - 2*x)*(1 - 26*x + 512*x^2)*(1 - 114*x - 1156*x^2 + 8*x^3)*(1 - 578*x - 228*x^2 + 8*x^3)). - Colin Barker, Feb 27 2018

A181195 Number of n X 8 matrices containing a permutation of 1..n*8 in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

Original entry on oeis.org

1, 429, 750325, 2152061145, 7372392431849, 26791156423752069, 99086837728795943389, 368253691037579094795185, 1370391973012218579798432209, 5101453317266742158652901045085

Offset: 1

R. H. Hardin, Oct 10 2010

nonn

Column 8 of A181196.

			Some solutions for 3 X 8:
..1..2..3..4..5..6..7..8....1..2..3..4..5..6..7.11....1..2..3..4..5..6..7.11
..9.10.11.12.13.14.15.16....8..9.10.12.13.16.17.19....8..9.10.12.13.16.17.20
.17.18.19.20.21.22.23.24...14.15.18.20.21.22.23.24...14.15.18.19.21.22.23.24

R. H. Hardin, Table of n, a(n) for n=1..100

Cf. A181196.

A181198 Number of 4 X n matrices containing a permutation of 1..4*n in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

Original entry on oeis.org

1, 1, 8, 169, 6392, 352184, 25097600, 2152061145, 212012802584, 23263015359672, 2781709560836960, 356806123331844056, 48516442013911012288, 6930091952294051922080, 1032505514388962439665280, 159544871422153344631037625, 25451354639006231998529405016

Offset: 1

R. H. Hardin, Oct 10 2010

nonn

			Some solutions for 4 X 4:
   1  2  3  4    1  2  3  4    1  2  3  4    1  2  3  4    1  2  3  4
   5  6  7  8    5  6  7  8    5  6  7  8    5  6  7  8    5  6  7  8
   9 10 11 12    9 10 11 13    9 10 11 14    9 10 12 13    9 10 12 14
  13 14 15 16   12 14 15 16   12 13 15 16   11 14 15 16   11 13 15 16

Christoph Koutschan, Table of n, a(n) for n = 1..100 (terms 1..27 from Alois P. Heinz)
Joerg Arndt, The a(4)=199 shifted standard Young tableaux of shape [4,4,4,4]
Manuel Kauers and Christoph Koutschan, Some D-finite and some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023, pp. 38-40.

Row 4 of A181196.

Table[
 NextPartitions[n1_, n2_, n3_, n4_] :=
   If[n1 < n, f[n1 + 1, n2, n3, n4], 0] +
   If[n2 < n1 - 1 || n2 === n - 1, f[n1, n2 + 1, n3, n4], 0] +
   If[n3 < n2 - 1 || n3 === n - 1 === n2 - 1, f[n1, n2, n3 + 1, n4], 0] +
   If[n4 < n3 - 1, f[n1, n2, n3, n4 + 1], 0];
 pp = f[1, 0, 0, 0];
 Do[pp = Expand[pp /. f[ns__] :> NextPartitions[ns]], {4 n - 2}];
 pp /. f[n, n, n, n - 1] -> 1,
{n, 20}] (* Christoph Koutschan, Feb 26 2023 *)

Conjectured recurrence of order 2 and degree 9: 3*(n + 1)*(2*n + 3)*(3*n + 4)*(3*n + 5)*(7*n^2 - 1)*(n + 2)^3*a(n + 2) - 8*(n + 1)*(2*n + 1)*(4*n + 3)*(4*n + 5)*(364*n^5 + 84*n^4 - 1025*n^3 - 534*n^2 + 157*n + 54)*a(n + 1) - 64*(2*n - 1)^2*(2*n + 1)*(4*n - 1)*(4*n + 1)*(4*n + 3)*(4*n + 5)*(7*n^2 + 14*n + 6)*a(n) = 0. - Christoph Koutschan, Feb 26 2023

Conjectured formula, solution to the above recurrence, for n > 1: a(n) = (-64)^n * (n-1) * (-1/2){2*n} * (1/2){n} / (4*(3*n)!) * (-1 + 3*Sum_{k=2..n-1} (-4)^k * (7*k^2-1) / ((k-1) * k * (k+1)^2 * (2*k-1)^2 * (2*k+1)^3) * binomial(3*k,2*k) * binomial(k+1/2,k)), where (a)_{n} is the Pochhammer symbol.

a(12)-a(27) from Alois P. Heinz, Jul 24 2012

a(28)-a(100) from Christoph Koutschan, Feb 26 2023

A181199 Number of 5 X n matrices containing a permutation of 1..5*n in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

Original entry on oeis.org

1, 1, 16, 985, 141696, 36372976, 14083834704, 7372392431849, 4848332563899256, 3808369342900073856, 3447336241467721584256, 3503140094024746011745456, 3918646197894288330216058576, 4753102567048482059557067412816, 6178133154985813161258658378449616

Offset: 1

R. H. Hardin, Oct 10 2010

nonn

			Some solutions for 5 X 4:
  1  2  3  4    1  2  3  4    1  2  3  4    1  2  3  4    1  2  3  4
  5  6  7  8    5  6  7  8    5  6  7  8    5  6  7  8    5  6  7  8
  9 10 11 12    9 10 11 12    9 10 11 12    9 10 11 12    9 10 11 12
 13 14 15 16   13 14 15 17   13 14 15 18   13 14 16 17   13 14 16 18
 17 18 19 20   16 18 19 20   16 17 19 20   15 18 19 20   15 17 19 20

Christoph Koutschan, Table of n, a(n) for n = 1..100 (terms 1..26 from Alois P. Heinz)
Manuel Kauers and Christoph Koutschan, Some D-finite and some Possibly D-finite Sequences in the OEIS, arXiv:2303.02793 [cs.SC], 2023, pp. 38-40.

Row 5 of A181196.

Table[
  NextPartitions[n1_, n2_, n3_, n4_, n5_] :=
    If[n1 < n, f[n1 + 1, n2, n3, n4, n5], 0] +
    If[n2 < n1 - 1 || n2 === n - 1, f[n1, n2 + 1, n3, n4, n5], 0] +
    If[n3 < n2 - 1 || n3 === n - 1 === n2 - 1, f[n1, n2, n3 + 1, n4, n5], 0] +
    If[n4 < n3 - 1 || n4 === n - 1 === n3 - 1, f[n1, n2, n3, n4 + 1, n5], 0] +
    If[n5 < n4 - 1, f[n1, n2, n3, n4, n5 + 1], 0];
  pp = f[1, 0, 0, 0, 0];
  Do[pp = Expand[pp /. f[ns__] :> NextPartitions[ns]], {5 n - 2}];
  pp /. f[n, n, n, n, n - 1] -> 1,
{n, 20}] (* Christoph Koutschan, Feb 27 2023 *)

Conjectured recurrence of order 3 and degree 24: 3*(n + 2)^2*(2*n + 3)*(2*n + 5)^2*(3*n + 7)*(3*n + 8)*(4*n + 9)*(4*n + 11)*(137855872*n^11 + 860969696*n^10 + 2047036856*n^9 + 2032587274*n^8 - 24192441*n^7 - 1894061166*n^6 - 1671661480*n^5 - 524330624*n^4 + 36004789*n^3 + 62751860*n^2 + 13865604*n + 927360)*(n + 3)^4*a(n + 3) - (n + 2)^2*(2*n + 3)*(717088749346816*n^21 + 14962008250398720*n^20 + 139665528127686656*n^19 + 760770273998231808*n^18 + 2616700630350746208*n^17 + 5556799141672247640*n^16 + 5462710847171622988*n^15 - 6080171043591548610*n^14 - 31771892184486994333*n^13 - 54791308745183340309*n^12 - 46634308960957698567*n^11 - 1215620047630796049*n^10 + 48833460779191449763*n^9 + 65692918122110754573*n^8 + 47097079083848116511*n^7 + 19628106098287909191*n^6 + 3499276436019940446*n^5 - 850259283025327524*n^4 - 685802053101717288*n^3 - 180089041888657440*n^2 - 22603977271411200*n - 1104708217536000)*a(n + 2) - 15*(2*n + 1)*(5*n + 6)*(5*n + 7)*(5*n + 8)*(5*n + 9)*(9113927409664*n^19 + 166164038596608*n^18 + 1357324488921088*n^17 + 6541981632511232*n^16 + 20586994844739808*n^15 + 44061606220301336*n^14 + 64302800289940820*n^13 + 61042725728660150*n^12 + 30740208891967721*n^11 - 3351043407516635*n^10 - 17410620603191989*n^9 - 12634296322883951*n^8 - 4487751704725767*n^7 - 793918042456325*n^6 - 54713722756179*n^5 + 42448206362469*n^4 + 50327965592874*n^3 + 21777752002716*n^2 + 4014999151560*n + 257403484800)*a(n + 1) + 25*(2*n - 1)^2*(2*n + 1)*(4*n - 1)*(4*n + 1)*(5*n + 1)*(5*n + 2)*(5*n + 3)*(5*n + 4)*(5*n + 6)*(5*n + 7)*(5*n + 8)*(5*n + 9)*(137855872*n^11 + 2377384288*n^10 + 18238806776*n^9 + 81945774178*n^8 + 238738633847*n^7 + 471293180347*n^6 + 638861237719*n^5 + 588363536007*n^4 + 354332674386*n^3 + 128320688700*n^2 + 23068181160*n + 1077753600)*a(n) = 0. - Christoph Koutschan, Feb 27 2023

Conjecture: a(n) ~ 9 * 5^(5*n + 1/2) / (2^17 * Pi^2 * n^12), based on the recurrence by Christoph Koutschan. - Vaclav Kotesovec, Feb 27 2023

a(11)-a(26) from Alois P. Heinz, Jul 24 2012

cp's OEIS Frontend

A181197 Number of 3 X n matrices containing a permutation of 1..3*n in increasing order rowwise, columnwise and (downwards) antidiagonally.

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A227578 Number A(n,k) of lattice paths from {n}^k to {0}^k using steps that decrement one component such that for each point (p_1,p_2,...,p_k) we have p_1<=p_2<=...<=p_k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A181191 Number of n X n matrices containing a permutation of 1..n*n in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

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Maple

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A181192 Number of n X 5 matrices containing a permutation of 1..n*5 in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

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A181193 Number of n X 6 matrices containing a permutation of 1..n*6 in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

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A181194 Number of n X 7 matrices containing a permutation of 1..n*7 in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

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A181195 Number of n X 8 matrices containing a permutation of 1..n*8 in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

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A181198 Number of 4 X n matrices containing a permutation of 1..4*n in increasing order rowwise, columnwise, diagonally and (downwards) antidiagonally.

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