A213978 -id:A213978 - cp's OEIS frontend

A214722 Number A(n,k) of solid standard Young tableaux of shape [[{n}^k],[n]]; square array A(n,k), n>=0, k>=1, read by antidiagonals.

1, 1, 1, 1, 2, 2, 1, 3, 16, 5, 1, 4, 91, 192, 14, 1, 5, 456, 5471, 2816, 42, 1, 6, 2145, 143164, 464836, 46592, 132, 1, 7, 9724, 3636776, 75965484, 48767805, 835584, 429, 1, 8, 43043, 91442364, 12753712037, 55824699632, 5900575762, 15876096, 1430

Offset: 0

Alois P. Heinz, Jul 26 2012

			Square array A(n,k) begins:
   1,     1,        1,           1,              1,                 1, ...
   1,     2,        3,           4,              5,                 6, ...
   2,    16,       91,         456,           2145,              9724, ...
   5,   192,     5471,      143164,        3636776,          91442364, ...
  14,  2816,   464836,    75965484,    12753712037,     2214110119572, ...
  42, 46592, 48767805, 55824699632, 70692556053053, 98002078234748974, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=1-4 give: A000108, A006335, A213978, A215220.
Rows n=0-3 give: A000012, A000027, A214824, A211505.
A(n,n) gives A258583.
Cf. A213932, A214637, A214631, A258586.

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

b[l_List] := b[l] = With[{m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]]] < j, 0, l[[i+1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]] ]; a[n_, k_] := b[{Array[n&, k], {n}}]; Table[Table[a[n, 1+d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 17 2013, translated from Maple *)

A214978 Array T(m,n) = Fibonacci(m*n)/Fibonacci(m), by antidiagonals; transpose of A028412.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 3, 8, 4, 1, 5, 21, 17, 7, 1, 8, 55, 72, 48, 11, 1, 13, 144, 305, 329, 122, 18, 1, 21, 377, 1292, 2255, 1353, 323, 29, 1, 34, 987, 5473, 15456, 15005, 5796, 842, 47, 1, 55, 2584, 23184, 105937, 166408, 104005, 24447, 2208, 76, 1, 89

Offset: 1

Clark Kimberling, Oct 27 2012

The main entry is the transpose, A028412. In the present format, the array can be compared directly with A214984 and A214986.

			Northwest corner:
  1  1   2    3      5       8
  1  3   8   21     55     144
  1  4  17   72    305    1292
  1  7  48  329   2255   15456
  1 11 122 1353  15005  166408
  1 18 323 5796 104005 1866294

Clark Kimberling, Antidiagonals n = 1..60, flattened

Cf. A000045, A028412, A214982, A214983, A214984, A214986.

F[n_] := Fibonacci[n]; t[m_, n_] := F[m*n]/F[m]
TableForm[Table[t[m, n], {m, 1, 10}, {n, 1, 10}]]
u = Table[t[k, n + 1 - k], {n, 1, 12}, {k, 1, n}];
v[n_] := Sum[F[m*(n + 1 - m)]/F[m], {m, 1, n}];
Flatten[u]                           (* A213978 *)
Flatten[Table[t[n, n], {n, 1, 20}]]  (* A051294 *)
Table[(t[n, 5] - 5)/50, {n, 1, 20}]  (* A214982 *)
Table[v[n], {n, 1, 30}]              (* A214983 *)

T(m,n) = Fibonacci(m*n)/Fibonacci(m).

A213932 Number of solid standard Young tableaux of shape [[n,n,n],[n,n]].

Original entry on oeis.org

1, 5, 707, 268326, 168146839, 143163177336, 149998192424502, 182598353781240533, 249032962712552804432, 371285830572997665257695, 594729699502746726969433566, 1010574132470951359396337494800, 1804193873947216124589237862262262

Offset: 0

Alois P. Heinz, Jul 23 2012

nonn

Also the number of solid standard Young tableaux of shape [[n,n],[n,n],[n]].

Alois P. Heinz, Table of n, a(n) for n = 0..30
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Cf. A213978, A214637, A214638.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
a:= n-> b([[n, n, n], [n, n]]):
seq(a(n), n=0..10);

b[l_] := b[l] = With[{ m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]]] < j, 0, l[[i+1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]]-1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]]]; a[n_] := b[{{n, n, n}, {n, n}}]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

A214986 Power ceiling array for the golden ratio, by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 8, 5, 1, 1, 12, 21, 22, 7, 1, 1, 20, 55, 94, 48, 12, 1, 1, 33, 144, 399, 329, 134, 18, 1, 1, 54, 377, 1691, 2255, 1487, 323, 30, 1, 1, 88, 987, 7164, 15456, 16492, 5796, 872, 47, 1, 1, 143, 2584, 30348, 105937, 182900

Offset: 1

Clark Kimberling, Oct 28 2012

row 0: A000012 ... row 6: A049660
row 1: A000071 ... row 8: A049668
row 2: A001906 ... col 0: A000012
row 3: A049652 ... col 1: A169986
row 4: A004187
For x>1, define c(x,0) = 1 and c(x,n) = ceiling(x*c(x,n-1)) for n>0.  Row m of A214986 is the sequence c(r^m,n), where r = golden ratio = (1 + sqrt(5))/2.  The name of the array corresponds to the power ceiling function f(x) = limit of c(x,n)/x^n as n increases without bound; f(x) generalizes the case for x = 3/2 as described under "Power Ceilings" at MathWorld.  For a graph of f(x), see the Mathematica program at A083286.
The term "power ceiling sequence" extends to sequences generated by recurrences P(n) = ceiling(x*P(n-1)) + g(n), and "power ceiling functions" f(x) to the limit of P(n)/x^n in case x>1 and g(n)/x^n -> 0.
Suppose that h is a nonnegative integer and g(n) is a constant.  If x is a positive integer power of the golden ratio r, then f(x), in many cases, lies in the field Q(sqrt(5)).  Examples matching rows of A214986, using g(n) = 0, follow:
...
x ... P ........ f(x)
r ... A000071 .. (5 + 2*sqrt(5))/2 = 1.8944... (A010532)
r^2 . A001906 .. (5 + 3*sqrt(5))/10 = 1.7082...(A176015)
r^3 . A049652 .. (25 + 11*sqrt(5))/40 = 1.2399...
r^4 . A004187 .. (15 + 7*sqrt(5))/10 = 1.0219...
...
If k is odd, then f(r^k) = r^k((b(k) + c(k))/d(k)), where
b(k) = L(j)^2 + L(j-1)^2, where j=[(k+1)/2], L=A000032 (Lucas numbers); c(k) = (L(k)+2)*sqrt(5); d(k) = 10*F(k)*L(k), where F=A000045 (Fibonacci numbers).  If k is even, then f(r^k) = r^k/(F(k)*sqrt(5)).

			Northwest corner:
1...1....1.....1......1.......1
1...2....4.....7......12......20
1...3....8.....21.....55......144
1...5....22....94.....399.....1691
1...7....48....329....2255....15456
1...19...134...1487...16492...182900

Clark Kimberling, Antidiagonals n = 1..35, flattened
Eric Weisstein's World of Mathematics, Power Ceilings

Cf. A000045, A214978, A214984, A214987.

r = GoldenRatio;
s[x_, 0] := 1; s[x_, n_] := Ceiling[x*s[x, n - 1]];
t = TableForm[Table[s[r^m, n], {m, 0, 10}, {n, 0, 10}]  ]
u = Flatten[Table[s[r^m, n - m], {n, 0, 10}, {m, 0, n}]]

The odd-numbered rows of A214986 are even-numbered rows of A213978; the even-numbered rows of A214986 are odd-numbered rows of A214984.

A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 4, 5, 1, 1, 10, 26, 10, 7, 1, 1, 28, 276, 258, 26, 11, 1, 1, 84, 3740, 14318, 3346, 76, 15, 1, 1, 264, 58604, 1161678, 1214358, 54108, 232, 22, 1, 1, 858, 1010616, 118316062, 741215012, 150910592, 1054256, 764, 30

Offset: 0

Alois P. Heinz, Aug 05 2012

			Square array A(n,k) begins:
:  1,  1,     1,         1,            1,                1, ...
:  1,  1,     1,         1,            1,                1, ...
:  2,  2,     4,        10,           28,               84, ...
:  3,  4,    26,       276,         3740,            58604, ...
:  5, 10,   258,     14318,      1161678,        118316062, ...
:  7, 26,  3346,   1214358,    741215012,     620383261034, ...
: 11, 76, 54108, 150910592, 840790914296, 7137345113624878, ...

Alois P. Heinz, Antidiagonals n = 0..15, flattened
S. B. Ekhad and D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=0-5 give: A000041, A000085, A215266, A290202, A290214, A290274.
Rows n=0+1, 2-5 give: A000012, 2*A000108, 2*A005789 + A006335, 2*A005790 + 2*A213978 + A114714, 2*A005791 + 2*A215220 + 2*A213932 + A214638.
Main diagonal gives A290225.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}minus{0}={}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
g:= proc(n, i, k, l) `if`(n=0 or i=1, b(map(x-> [k$x], [l[], 1$n])),
       add(g(n-i*j, i-1, k, [l[], i$j]), j=0..n/i))
    end:
A:= (n, k)-> g(n, n, k, []):
seq(seq(A(n, d-n), n=0..d), d=0..10);

b[l_] := b[l] = With[{m = Length[l]}, If[Union[l // Flatten] ~Complement~ {0} == {}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i + 1]]] < j, 0, l[[i + 1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j + 1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] - 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]]];
g[n_, i_, k_, l_] := If[n == 0 || i == 1, b[Table[k, {#}]& /@ Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, k, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
A[n_, k_] := g[n, n, k, {}];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Sep 24 2022, after Alois P. Heinz *)

A214637 Number of solid standard Young tableaux of shape [[n,n,n],[n,n],[n]].

Original entry on oeis.org

1, 16, 17086, 61189172, 404233159860, 3880365678824980, 47959061464818182058, 711513280222442751394224, 12121127323153614807021655742, 230127245538294682127207785787376, 4767460278053986542112719904243778834, 106115342273795146740243750912097789131600

Offset: 0

Alois P. Heinz, Jul 23 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..25
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Cf. A213932, A213978, A214638.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
a:= n-> b([[n, n, n], [n, n], [n]]):
seq(a(n), n=0..10);

b[l_] := b[l] = With[{m := Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]]] < j, 0, l[[i+1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]]-1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]] ]; a[n_] := b[{{n, n, n}, {n, n}, {n}}]; Table[a[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

A214638 Number of solid standard Young tableaux of shape [[n,n,n],[n],[n]].

Original entry on oeis.org

1, 6, 936, 379366, 249664758, 221005209058, 239143562020194, 299233941746052998, 417999868371999142276, 636568066798406010872120, 1039267652960081699025215774, 1796704965351078502372895796786, 3258764657213579008313421745034602

Offset: 0

Alois P. Heinz, Jul 23 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..40
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Cf. A213932, A213978, A214637.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
a:= n-> b([[n, n, n], [n], [n]]):
seq(a(n), n=0..10);

b[l_] := b[l] = With[{m := Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]]] < j, 0, l[[i+1, j]]] && l[[i, j]] > If[Length[l[[i]]] == j, 0, l[[i, j+1]]], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]]-1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, m}]] ]; a[n_] := b[{{n, n, n}, {n}, {n}}]; Table[a[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

A214631 Number A(n,k) of solid standard Young tableaux of shape [[(n)^(k+1)],[n]^k]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 16, 1, 1, 20, 936, 192, 1, 1, 70, 85800, 379366, 2816, 1, 1, 252, 9962680, 1825221320, 249664758, 46592, 1, 1, 924, 1340103744, 14336196893200, 89261675900020, 221005209058, 835584, 1

Offset: 0

Alois P. Heinz, Jul 26 2012

			Square array A(n,k) begins:
  1,    1,         1,              1,                    1, ...
  1,    2,         6,             20,                   70, ...
  1,   16,       936,          85800,              9962680, ...
  1,  192,    379366,     1825221320,       14336196893200, ...
  1, 2816, 249664758, 89261675900020, 70351928759681296000, ...

Alois P. Heinz, Antidiagonals n = 0..12
S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012
Wikipedia, Young tableau

Columns k=0-2 give: A000012, A006335, A214638.
Rows n=0-1 give: A000012, A000984.
Cf. A213932, A213978, A214637, A214722.

b:= proc(l) option remember; local m; m:= nops(l);
      `if`({map(x-> x[], l)[]}={0}, 1, add(add(`if`(l[i][j]>
      `if`(i=m or nops(l[i+1])
      `if`(nops(l[i])=j, 0, l[i][j+1]), b(subsop(i=subsop(
       j=l[i][j]-1, l[i]), l)), 0), j=1..nops(l[i])), i=1..m))
    end:
A:= (n, k)-> b([[n$(k+1)], [n]$k]):
seq(seq(A(n, d-n), n=0..d), d=0..8);

b[l_] := b[l] = With[{m = Length[l]}, If[Union[Flatten[l]] == {0}, 1, Sum[Sum[If[l[[i, j]] > If[i == m || Length[l[[i+1]] ] < j, 0, l[[i+1, j]] ] && l[[i, j]] > If[Length[l[[i]] ] == j, 0, l[[i, j+1]] ], b[ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]]-1]]], 0], {j, 1, Length[l[[i]] ]}], {i, 1, m}]]]; a[n_, k_] := b[{Array[n&, k+1], Sequence @@ Array[{n}&, k]}]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 8}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)

A214987 Power round array for the golden ratio, by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 8, 4, 1, 1, 8, 21, 17, 7, 1, 1, 13, 55, 72, 48, 11, 1, 1, 21, 144, 305, 329, 122, 18, 1, 1, 34, 377, 1292, 2255, 1353, 323, 29, 1, 1, 55, 987, 5473, 15456, 15005, 5796, 842, 47, 1, 1, 89, 2584, 23184, 105937, 166408, 104005

Offset: 1

Clark Kimberling, Oct 28 2012

The term "power round sequence" (after "power ceiling sequence" at A214986) extends to sequences generated by recurrences P(n) = round(x*P(n-1)) + g(n), and "power round functions" f(x) to the limit of P(n)/x^n in case x>1 and g(n)/x^n -> 0.  Suppose that h is a nonnegative integer and g(n) is a constant.  If x is a positive integer power of the golden ratio r, then f(x), in many cases, lies in the field Q(sqrt(5)).  Examples matching rows of A214987, using g(n) = 0, follow:
...
x ... P . .. . . f(x)
r ... A000045 .. 1/2 + 3*sqrt(5)/10 = 1.1708... (A176015)
r^2 . A001906 .. 1/2 + 3*sqrt(5)/10 = 1.1708... (A176015)
r^3 . A001076 .. 1/2 + sqrt(5)/5 = 0.9472...
r^4 . A004187 .. 1/2 + 7*sqrt(5)/30 = 1.0217...
In general, f(r^k) = 1/2 + sqrt(5)*L(k)/(10*F(k)) for k>1, where L = A000032 (Lucas numbers) and F = A000045 (Fibonacci numbers).
(row 2 of A214987) = (row 1 of A213978 except for its initial 1)
(row n of A214987) = (row n-1 of A213978 for n>2).

			1...1...1....1.....1......1
1...2...3....5.....8......13
1...3...8....21....5......144
1...4...17...72....305....1292
1...7...48...329...2255...15456

Clark Kimberling, Antidiagonals n = 1..35, flattened

Cf. A000045, A214978, A214984, A214986.

r = GoldenRatio;
s[x_, 0] := 1; s[x_, n_] := Round[x*s[x, n - 1]];
t = TableForm[Table[s[r^m, n], {m, 0, 10}, {n, 0, 10}]  ]
u = Flatten[Table[s[r^m, n - m], {n, 0, 10}, {m, 0, n}]]

cp's OEIS Frontend

A214722 Number A(n,k) of solid standard Young tableaux of shape [[{n}^k],[n]]; square array A(n,k), n>=0, k>=1, read by antidiagonals.

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A214978 Array T(m,n) = Fibonacci(m*n)/Fibonacci(m), by antidiagonals; transpose of A028412.

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A213932 Number of solid standard Young tableaux of shape [[n,n,n],[n,n]].

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A214986 Power ceiling array for the golden ratio, by antidiagonals.

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A215204 Number A(n,k) of solid standard Young tableaux of cylindrical shape lambda X k, where lambda ranges over all partitions of n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214637 Number of solid standard Young tableaux of shape [[n,n,n],[n,n],[n]].

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A214638 Number of solid standard Young tableaux of shape [[n,n,n],[n],[n]].

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Mathematica

A214631 Number A(n,k) of solid standard Young tableaux of shape [[(n)^(k+1)],[n]^k]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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