A207981 -id:A207981 - cp's OEIS frontend

A207868 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

1, 1, 1, 2, 4, 2, 5, 34, 34, 5, 15, 500, 2052, 500, 15, 52, 10900, 278982, 278982, 10900, 52, 203, 322768, 68162042, 455546040, 68162042, 322768, 203, 877, 12297768, 26419793726, 1625686993918, 1625686993918, 26419793726, 12297768, 877

Offset: 1

R. H. Hardin, Feb 21 2012

Table starts
...1.........1..............2.................5.................15
...1.........4.............34...............500..............10900
...2........34...........2052............278982...........68162042
...5.......500.........278982.........455546040......1625686993918
..15.....10900.......68162042.....1625686993918.103204230192540988
..52....322768....26419793726.10764437129618296
.203..12297768.15002771641712
.877.580849872

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

Andrew Howroyd, Table of n, a(n) for n = 1..231 (terms 1..49 from R. H. Hardin)
Eric Weisstein's World of Mathematics, Grid Graph.
Eric Weisstein's World of Mathematics, Vertex Coloring.
Wikipedia, Graph Coloring.

Columns 1..5 are A000110(n-1), A207864, A207865, A207866, A207867.
Main diagonal is A207863.
Cf. A207997 (3 colorings), A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings).
Cf. A207981, A208001 (knight), A208021 (king), A208054, A208096, A208301.

A207978 Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 2, 7, 67, 1080, 25287, 794545, 31858034, 1573857867, 93345011951, 6514819011216, 526593974392123, 48658721593531669, 5084549201524804642, 595348294459678745663, 77500341343460209843627, 11140107960738185817545800, 1757660562895916320583653791

Offset: 0

R. H. Hardin, Feb 22 2012

nonn

Column 2 of A207981.
a(n) is equal to the number of set partitions of {1,2,...,2n} such that k and k+2 do not appear in the same block for any k. - Andrew Howroyd , May 23 2023
a(n) is equal to the number of set partitions of {1,2,...,2n} such that the only sets of size 1 in the set partition are either {1} or {2}.
a(n) is also the dimension of the centralizer algebra End_{S_m}((V^{(m-1,1)}{S_m})^{\otimes n-1} \otimes V_m ) where V^{(m-1,1)}{S_m} is an irreducible S_m module indexed by (m-1,1) and V_m is the permutation module for S_m (with the condition that m is sufficiently large). - Mike Zabrocki, May 23 2023

			Some solutions for n=4:
..0..0....0..0....0..0....0..0....0..0....0..0....0..1....0..0....0..0....0..1
..1..1....1..1....1..1....1..1....1..1....1..1....0..1....1..1....1..1....2..1
..2..3....2..0....2..0....2..2....0..2....0..0....0..1....0..2....0..2....0..1
..2..0....3..3....1..0....3..4....3..1....1..1....0..1....3..4....0..1....0..1
The set partitions of 4 where at most {1} and {2} are the only sets of size 1 are {1234}, {1|234}, {2|134}, {12|34}, {13|24}, {14|23}, {1|2|34} - _Mike Zabrocki_, May 23 2023

Alois P. Heinz, Table of n, a(n) for n = 0..288 (terms n = 1..40 from R. H. Hardin)
Rosa Orellana, Nancy Wallace, and Mike Zabrocki, Representations of the quasi-partition algebras, arXiv:2306.17326 [math.RT], 2023.

Cf. A000110, A011968, A207981.

a:=n->add((-1)^s*binomial(2*n-2, s) * combinat[bell](2*n-s), s = 0 .. 2*n); # Mike Zabrocki, May 23 2023
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> `if`(n=0, 1, b(2*n-1)+b(2*n-2)):
seq(a(n), n=0..19);  # Alois P. Heinz, May 30 2023

b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j] Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := If[n == 0, 1, b[2n-1] + b[2n-2]];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 17 2024, after Alois P. Heinz *)

a = lambda n: sum((-1)**s*binomial(2*n-2,s)*bell_number(2*n-s) for s in range(2*n-2+1)) # Mike Zabrocki, May 23 2023

a(n) = Sum_{s=0..2n} (-1)^s binomial(2n-2,s) Bell(2n-s). - Mike Zabrocki, May 23 2023

a(n) = A011968(2*n-1) for n>=1. - Alois P. Heinz, May 30 2023

New description and a formula added by Mike Zabrocki, May 23 2023

a(0)=1 prepended by Alois P. Heinz, May 30 2023

A207979 Number of n X 3 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

5, 67, 4192, 587208, 147512732, 58668937932, 34119101220224, 27451586022436696, 29334574202437277520, 40343697156563884675488, 69646199705725564208484736, 147875840329637502384856552096, 379682313150983568999164195148800

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..190 (terms 1..16 from R. H. Hardin)

Column 3 of A207981.

A207980 Number of n X 4 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

15, 1080, 587208, 953124784, 3410310590912, 22727131144960768, 253335104721267417984, 4382417290595586402531584, 111319355727036703898865465344, 3978834602846298450265688566196224, 193442461762324413798952526604793292800

Offset: 1

R. H. Hardin, Feb 22 2012

nonn

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

Andrew Howroyd, Table of n, a(n) for n = 1..140

Column 4 of A207981.

Terms a(8) and beyond from Andrew Howroyd, Mar 15 2023

cp's OEIS Frontend

A207868 T(n,k)=Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A207978 Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

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A207979 Number of n X 3 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

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A207980 Number of n X 4 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

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A361452 Number of colorings of an n X n grid up to permutation of the colors with no element having the same color as any diagonal or antidiagonal neighbor.

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