A049400 -id:A049400 - cp's OEIS frontend

A182172 Number A(n,k) of standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 6, 1, 0, 1, 1, 2, 4, 9, 10, 1, 0, 1, 1, 2, 4, 10, 21, 20, 1, 0, 1, 1, 2, 4, 10, 25, 51, 35, 1, 0, 1, 1, 2, 4, 10, 26, 70, 127, 70, 1, 0, 1, 1, 2, 4, 10, 26, 75, 196, 323, 126, 1, 0, 1, 1, 2, 4, 10, 26, 76, 225, 588, 835, 252, 1, 0

Offset: 0

Alois P. Heinz, Apr 16 2012

Also the number A(n,k) of standard Young tableaux of n cells and <= k columns.

A(n,k) is also the number of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,ak), where #(z,x) counts the letters x in word z. The A(4,4) = 10 words of length 4 over alphabet {a,b,c,d} are: aaaa, aaab, aaba, abaa, aabb, abab, aabc, abac, abca, abcd.

			A(4,2) = 6, there are 6 standard Young tableaux of 4 cells and height <= 2:
  +------+  +------+  +---------+  +---------+  +---------+  +------------+
  | 1  3 |  | 1  2 |  | 1  3  4 |  | 1  2  4 |  | 1  2  3 |  | 1  2  3  4 |
  | 2  4 |  | 3  4 |  | 2 .-----+  | 3 .-----+  | 4 .-----+  +------------+
  +------+  +------+  +---+        +---+        +---+
Square array A(n,k) begins:
  1,  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,  2,   2,   2,   2,   2,   2,   2, ...
  0,  1,  3,   4,   4,   4,   4,   4,   4, ...
  0,  1,  6,   9,  10,  10,  10,  10,  10, ...
  0,  1, 10,  21,  25,  26,  26,  26,  26, ...
  0,  1, 20,  51,  70,  75,  76,  76,  76, ...
  0,  1, 35, 127, 196, 225, 231, 232, 232, ...
  0,  1, 70, 323, 588, 715, 756, 763, 764, ...

Alois P. Heinz, Antidiagonals n = 0..80, flattened
Wikipedia, Young tableau

Columns k=0-12 give: A000007, A000012, A001405, A001006, A005817, A049401, A007579, A007578, A007580, A212915, A212916, A229053, A229068.
Main diagonal gives A000085.
A(2n,n) gives A293128.
Cf. A047884, A049400, A226873, A240608.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
       +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) option remember;
      `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
A:= (n, k)-> g(n, k, []):
seq(seq(A(n, d-n), n=0..d), d=0..15);

h[l_List] := Module[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_List] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Array[1&, n]]], g [n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]]];
a[n_, k_] := g[n, k, {}];
Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 06 2013, translated from Maple *)

Conjecture: A(n,k) ~ k^n/Pi^(k/2) * (k/n)^(k*(k-1)/4) * Product_{j=1..k} Gamma(j/2). - Vaclav Kotesovec, Sep 12 2013

A047884 Triangle of numbers a(n,k) = number of Young tableaux with n cells and k rows (1 <= k <= n); also number of self-inverse permutations on n letters in which the length of the longest scattered (i.e., not necessarily contiguous) increasing subsequence is k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 9, 11, 4, 1, 1, 19, 31, 19, 5, 1, 1, 34, 92, 69, 29, 6, 1, 1, 69, 253, 265, 127, 41, 7, 1, 1, 125, 709, 929, 583, 209, 55, 8, 1, 1, 251, 1936, 3356, 2446, 1106, 319, 71, 9, 1, 1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10, 1

Offset: 1

Wouter Meeussen

			For n=3 the 4 tableaux are
  1 2 3 . 1 2 . 1 3 . 1
  . . . . 3 . . 2 . . 2
  . . . . . . . . . . 3
Triangle begins:
  1;
  1,   1;
  1,   2,    1;
  1,   5,    3,     1;
  1,   9,   11,     4,     1;
  1,  19,   31,    19,     5,    1;
  1,  34,   92,    69,    29,    6,    1;
  1,  69,  253,   265,   127,   41,    7,   1;
  1, 125,  709,   929,   583,  209,   55,   8,  1;
  1, 251, 1936,  3356,  2446, 1106,  319,  71,  9,  1;
  1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10,  1;
  ...

W. Fulton, Young Tableaux, Cambridge, 1997.
D. Stanton and D. White, Constructive Combinatorics, Springer, 1986.

Alois P. Heinz, Rows n = 1..68, flattened
R. P. Stanley, A combinatorial miscellany
Index entries for sequences related to Young tableaux.

Row sums give A000085.
Cf. A049400, A049401, and A178249 which imposes contiguity.
Columns k=1-10 give: A000012, A014495, A217323, A217324, A217325, A217326, A217327, A217328, A217321, A217322. - Alois P. Heinz, Oct 03 2012
a(2n,n) gives A267436.

h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, (p->h(p)*x^`if`(p=[], 0, p[1]))
      ([l[], 1$n]), add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2, [])):
seq(T(n), n=1..14); # Alois P. Heinz, Apr 16 2012, revised Mar 05 2014

Table[ Plus@@( NumberOfTableaux/@ Reverse/@Union[ Sort/@(Compositions[ n-m, m ]+1) ]), {n, 12}, {m, n} ]
(* Second program: *)
h[l_] := With[{n=Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+Sum[If[ l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n== 0|| i==1, Function[p, h[p]*x^If[p == {}, 0, p[[1]] ] ] [ Join[l, Array[1&, n]]], Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n, {}]];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Oct 26 2015, after Alois P. Heinz *)

Definition amended ('scattered' added) by Wouter Meeussen, Dec 22 2010

A182222 Number T(n,k) of standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 10, 10, 9, 4, 1, 26, 26, 25, 16, 5, 1, 76, 76, 75, 56, 25, 6, 1, 232, 232, 231, 197, 105, 36, 7, 1, 764, 764, 763, 694, 441, 176, 49, 8, 1, 2620, 2620, 2619, 2494, 1785, 856, 273, 64, 9, 1, 9496, 9496, 9495, 9244, 7308, 3952, 1506, 400, 81, 10, 1

Offset: 0

Alois P. Heinz, Apr 19 2012

Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= k. T(4,3) = 4: 1234, 1243, 1324, 2134; T(3,0) = T(3,1) = 4: 123, 132, 213, 321; T(5,3) = 16: 12345, 12354, 12435, 12543, 13245, 13254, 14325, 14523, 15342, 21345, 21354, 21435, 32145, 34125, 42315, 52341.

			T(4,3) = 4, there are 4 standard Young tableaux of 4 cells and height >= 3:
  +---+   +------+   +------+   +------+
  | 1 |   | 1  2 |   | 1  3 |   | 1  4 |
  | 2 |   | 3 .--+   | 2 .--+   | 2 .--+
  | 3 |   | 4 |      | 4 |      | 3 |
  | 4 |   +---+      +---+      +---+
  +---+
Triangle T(n,k) begins:
    1;
    1,   1;
    2,   2,   1;
    4,   4,   3,   1;
   10,  10,   9,   4,   1;
   26,  26,  25,  16,   5,   1;
   76,  76,  75,  56,  25,   6,  1;
  232, 232, 231, 197, 105,  36,  7,  1;
  764, 764, 763, 694, 441, 176, 49,  8,  1;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Involution (mathematics)
Wikipedia, Young tableau

Columns 0-10 give: A000085, A000085 (for n>0), A001189, A218263, A218264, A218265, A218266, A218267, A218268, A218269, A218262.
Diagonal and lower diagonals give: A000012, A000027(n+1), A000290(n+1) for n>0, A131423(n+1) for n>1.
T(2n,n) gives A318289.
Cf. A047884, A049400, A182172.

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) option remember;
      `if`(n=0, h(l), `if`(i<1, 0, `if`(i=1, h([l[], 1$n]),
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
T:= (n, k)-> g(n, n, []) -`if`(k=0, 0, g(n, k-1, [])):
seq(seq(T(n, k), k=0..n), n=0..12);

h[l_] := Module[{n = Length[l]}, Sum[i, {i, l}]! / Product[ Product[1 + l[[i]] - j + Sum [If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Array[1&, n]]], g [n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]]];
t[n_, k_] := g[n, n, {}] - If[k == 0, 0, g[n, k-1, {}]];
Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

T(n,k) = A182172(n,n) - A182172(n,k-1) for k>0, T(n,0) = A182172(n,n).

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A182172 Number A(n,k) of standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A047884 Triangle of numbers a(n,k) = number of Young tableaux with n cells and k rows (1 <= k <= n); also number of self-inverse permutations on n letters in which the length of the longest scattered (i.e., not necessarily contiguous) increasing subsequence is k.

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A182222 Number T(n,k) of standard Young tableaux of n cells and height >= k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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