A182172 -id:A182172 - cp's OEIS frontend

A293109 Number T(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 5, 10, 4, 1, 0, 7, 24, 17, 5, 1, 0, 11, 62, 58, 26, 6, 1, 0, 15, 140, 193, 107, 37, 7, 1, 0, 22, 329, 603, 439, 178, 50, 8, 1, 0, 30, 725, 1852, 1663, 852, 275, 65, 9, 1, 0, 42, 1631, 5539, 6283, 3767, 1500, 402, 82, 10, 1

Offset: 0

Alois P. Heinz, Sep 30 2017

			Triangle T(n,k) begins:
  1;
  0,  1;
  0,  2,   1;
  0,  3,   3,   1;
  0,  5,  10,   4,   1;
  0,  7,  24,  17,   5,   1;
  0, 11,  62,  58,  26,   6,  1;
  0, 15, 140, 193, 107,  37,  7, 1;
  0, 22, 329, 603, 439, 178, 50, 8, 1;

Alois P. Heinz, Rows n = 0..40, flattened

Columns k=0-10 give: A000007, A000041 (for n>0), A293797, A293798, A293799, A293800, A293801, A293802, A293803, A293804, A293805.
Row sums give A293110.
T(2n,n) gives A293111.
Cf. A182172, A293108, A293113.

h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
    n, 0, g(n-i, i, [l[], i])))))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(g(d, k, [])
      *d, d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);

h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] < j, 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, n}]][Length[l]];
g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Table[1, n]]], g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Append[l, i]]]]]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[g[d, k, {}]*d, {d, Divisors[j]}]*A[n - j, k], {j, 1, n}]/n];
T[n_, 0] := A[n, 0]; T[n_, k_] := A[n, k] - A[n, k - 1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 09 2018, after Alois P. Heinz *)

T(n,k) = A293108(n,k) - A293108(n,k-1) for k>0, T(n,0) = A293108(n,0).

A293113 Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 2, 8, 4, 1, 0, 3, 20, 16, 5, 1, 0, 4, 47, 53, 25, 6, 1, 0, 5, 106, 173, 102, 36, 7, 1, 0, 6, 237, 532, 410, 172, 49, 8, 1, 0, 8, 522, 1615, 1545, 813, 268, 64, 9, 1, 0, 10, 1146, 4785, 5784, 3576, 1448, 394, 81, 10, 1

Offset: 0

Alois P. Heinz, Sep 30 2017

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,   1;
  0, 2,   3,   1;
  0, 2,   8,   4,   1;
  0, 3,  20,  16,   5,   1;
  0, 4,  47,  53,  25,   6,  1;
  0, 5, 106, 173, 102,  36,  7, 1;
  0, 6, 237, 532, 410, 172, 49, 8, 1;
  ...

Alois P. Heinz, Rows n = 0..40, flattened

Columns k=0-10 give: A000007, A000009 (for n>0), A293883, A293884, A293885, A293886, A293887, A293888, A293889, A293890, A293891.
Row sums give A293114.
T(2n,n) gives A293115.
Cf. A182172, A293109, A293112.

h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
    n, 0, g(n-i, i, [l[], i])))))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(g(i, k, []), j), j=0..n/i)))
    end:
T:= (n, k)-> b(n$2, k)-`if`(k=0, 0, b(n$2, k-1)):
seq(seq(T(n, k), k=0..n), n=0..14);

h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] n, 0, g[n - i, i, Append[l, i]]]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n - i*j, i-1, k]*Binomial[g[i, k, {}], j], {j, 0, n/i}]]];
T[n_, k_] := b[n, n, k] - If[k == 0, 0, b[n, n, k - 1]];
Table[T[n, k], {n, 0, 14}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

T(n,k) = A293112(n,k) - A293112(n,k-1) for k>0, T(n,0) = A293112(n,0).

A293108 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 3, 3, 0, 1, 1, 3, 6, 5, 0, 1, 1, 3, 7, 15, 7, 0, 1, 1, 3, 7, 19, 31, 11, 0, 1, 1, 3, 7, 20, 48, 73, 15, 0, 1, 1, 3, 7, 20, 53, 131, 155, 22, 0, 1, 1, 3, 7, 20, 54, 157, 348, 351, 30, 0, 1, 1, 3, 7, 20, 54, 163, 455, 954, 755, 42, 0

Offset: 0

Alois P. Heinz, Sep 30 2017

			Square array A(n,k) begins:
  1,  1,   1,   1,   1,   1,   1,   1, ...
  0,  1,   1,   1,   1,   1,   1,   1, ...
  0,  2,   3,   3,   3,   3,   3,   3, ...
  0,  3,   6,   7,   7,   7,   7,   7, ...
  0,  5,  15,  19,  20,  20,  20,  20, ...
  0,  7,  31,  48,  53,  54,  54,  54, ...
  0, 11,  73, 131, 157, 163, 164, 164, ...
  0, 15, 155, 348, 455, 492, 499, 500, ...

Alois P. Heinz, Antidiagonals n = 0..80, flattened

Columns k=0-10 give: A000007, A000041, A293732, A293733, A293734, A293735, A293736, A293737, A293738, A293739, A293740.
Main diagonal gives A293110.
Cf. A182172, A293109, A293112.

h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
    n, 0, g(n-i, i, [l[], i])))))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1, add(add(g(d, k, [])
      *d, d=numtheory[divisors](j))*A(n-j, k), j=1..n)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

h[l_] := Function [n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] < j, 0, 1], {k, i+1, n}], {j, 1, l[[i]]}], {i, n}]][Length[l]];
g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Table[1, n]]], g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Append[l, i]]]]]];
A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[Sum[g[d, k, {}]*d, {d, Divisors[j] }]*A[n - j, k], {j, 1, n}]/n];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

G.f. of column k: Product_{j>=1} 1/(1-x^j)^A182172(j,k).
A(n,k) = Sum_{j=0..k} A293109(n,j).
A(n,n) = A(n,k) for all k >= n.

A293112 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 5, 2, 0, 1, 1, 2, 6, 10, 3, 0, 1, 1, 2, 6, 14, 23, 4, 0, 1, 1, 2, 6, 15, 39, 51, 5, 0, 1, 1, 2, 6, 15, 44, 104, 111, 6, 0, 1, 1, 2, 6, 15, 45, 129, 284, 243, 8, 0, 1, 1, 2, 6, 15, 45, 135, 386, 775, 530, 10, 0

Offset: 0

Alois P. Heinz, Sep 30 2017

			Square array A(n,k) begins:
  1, 1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   1,   1,   1,   1,   1,   1, ...
  0, 1,   2,   2,   2,   2,   2,   2, ...
  0, 2,   5,   6,   6,   6,   6,   6, ...
  0, 2,  10,  14,  15,  15,  15,  15, ...
  0, 3,  23,  39,  44,  45,  45,  45, ...
  0, 4,  51, 104, 129, 135, 136, 136, ...
  0, 5, 111, 284, 386, 422, 429, 430, ...

Alois P. Heinz, Antidiagonals n = 0..80, flattened

Columns k=0-10 give: A000007, A000009, A293741, A293742, A293743, A293744, A293745, A293746, A293747, A293748, A293749.
Main diagonal gives A293114.
Cf. A182172, A293108, A293113.

h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
    n, 0, g(n-i, i, [l[], i])))))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*binomial(g(i, k, []), j), j=0..n/i)))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);

h[l_] := Function[n, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[ l[[k]] n, 0, g[n - i, i, Append[l, i]]]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*Binomial[g[i, k, {}], j], {j, 0, n/i}]]];
A[n_, k_] := b[n, n, k];
Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jun 03 2018, from Maple *)

G.f. of column k: Product_{j>=1} (1+x^j)^A182172(j,k).

A(n,k) = Sum_{j=0..k} A293113(n,j).

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).

A240608 Number A(n,k) of n-length words w over a k-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 4, 1, 0, 1, 1, 2, 5, 7, 1, 0, 1, 1, 2, 5, 13, 14, 1, 0, 1, 1, 2, 5, 14, 35, 25, 1, 0, 1, 1, 2, 5, 14, 45, 94, 50, 1, 0, 1, 1, 2, 5, 14, 46, 149, 254, 91, 1, 0, 1, 1, 2, 5, 14, 46, 164, 509, 688, 182, 1, 0, 1, 1, 2, 5, 14, 46, 165, 629, 1756, 1872, 336, 1, 0

Offset: 0

Alois P. Heinz, Apr 09 2014

			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,  4,   5,    5,    5,    5,    5,    5, ...
  0, 1,  7,  13,   14,   14,   14,   14,   14, ...
  0, 1, 14,  35,   45,   46,   46,   46,   46, ...
  0, 1, 25,  94,  149,  164,  165,  165,  165, ...
  0, 1, 50, 254,  509,  629,  650,  651,  651, ...
  0, 1, 91, 688, 1756, 2511, 2742, 2770, 2771, ...

Alois P. Heinz, Antidiagonals n = 0..36, flattened

Columns k=0-10 give: A000007, A000012, A026010(n-1) for n>0, A240609, A240610, A240611, A240612, A240613, A240614, A240615, A240616.
Main diagonal gives A240617.
Cf. A182172.

b:= proc(n, k, l) option remember; `if`(n=0, 1, `if`(nops(l) b(n, min(k, n), []):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, k_, l_List] := b[n, k, l] = If[n == 0, 1, If[Length[l] l[[i]]+1]], 0], {i, 1, Length[l]}]]; A[n_, k_] := b[n, Min[k, n], {}]; Table[ Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

A293110 Number of multisets of nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

Original entry on oeis.org

1, 1, 3, 7, 20, 54, 164, 500, 1630, 5472, 19257, 70133, 265858, 1042346, 4235031, 17760943, 76913277, 342919431, 1573637985, 7415371293, 35860511131, 177641956111, 900782461170, 4668600610346, 24714284921937, 133467868645017, 734844788634269, 4120752558254581

Offset: 0

Alois P. Heinz, Sep 30 2017

nonn

			a(0) = 1: {}.
a(1) = 1: {a}
a(2) = 3: {a,a}, {aa}, {ab}.
a(3) = 7: {a,a,a}, {a,aa}, {a,ab}, {aaa}, {aab}, {aba}, {abc}.

Alois P. Heinz, Table of n, a(n) for n = 0..800

Main diagonal of A293108.
Row sums of A293109 and of A293808.
Cf. A000085, A182172, A293114.

g:= proc(n) option remember;
      `if`(n<2, 1, g(n-1)+(n-1)*g(n-2))
    end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(g(d)
      *d, d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);

g[n_] := g[n] = If[n < 2, 1, g[n - 1] + (n - 1)*g[n - 2]];
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[g[d]*d, {d, Divisors[j]}]*a[n - j], {j, 1, n}]/n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 07 2018, from Maple *)

G.f.: Product_{j>=1} 1/(1-x^j)^A000085(j).

A007580 Number of Young tableaux of height <= 8.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 76, 232, 764, 2619, 9486, 35596, 139392, 562848, 2352064, 10092160, 44546320, 201158620, 930213752, 4387327088, 21115314916, 103386386516, 515097746072, 2605341147472, 13378787264584, 69622529312665, 367161088308490, 1959294979429380

Offset: 0

Simon Plouffe

nonn

Also the number of n-length words w over 8-ary alphabet {a1,a2,...,a8} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a8), where #(z,x) counts the letters x in word z. - Alois P. Heinz, May 30 2012

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Preprint. (Annotated scanned copy)
F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
Index entries for sequences related to Young tableaux.

Column k=8 of A182172. - Alois P. Heinz, May 30 2012

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, h([l[], 1$n]), `if`(i<1, 0,
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
a:= n-> g(n, 8, []):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 10 2012
# second Maple program:
a:= proc(n) option remember;
      `if`(n<4, [1, 1, 2, 4][n+1],
       ((40*n^3+1084*n^2+8684*n+18480)*a(n-1)
       +16*(n-1)*(5*n^3+107*n^2+610*n+600)*a(n-2)
       -1024*(n-1)*(n-2)*(n+6)*a(n-3)
       -1024*(n-1)*(n-2)*(n-3)*(n+4)*a(n-4)) /
       ((n+7)*(n+12)*(n+15)*(n+16)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

RecurrenceTable[{1024 (-3+n) (-2+n) (-1+n) (4+n) a[-4+n]+1024 (-2+n) (-1+n) (6+n) a[-3+n]-16 (-1+n) (600+610 n+107 n^2+5 n^3) a[-2+n]-4 (4620+2171 n+271 n^2+10 n^3) a[-1+n]+(7+n) (12+n) (15+n) (16+n) a[n]==0,a[1]==1,a[2]==2,a[3]==4,a[4]==10}, a, {n, 20}] (* Vaclav Kotesovec, Sep 11 2013 *)

a(n) ~ 135/16 * 8^(n+14)/(Pi^2*n^14). - Vaclav Kotesovec, Sep 11 2013

More terms from Alois P. Heinz, Apr 10 2012

A212915 Number of standard Young tableaux of n cells and height <= 9.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9495, 35685, 140031, 567503, 2382394, 10290308, 45780063, 208852719, 977152266, 4674398032, 22854255698, 113957313538, 579157509082, 2995214721530, 15752586526189, 84145056172981, 456221504976506, 2508227921637772

Offset: 0

Alois P. Heinz, May 30 2012

nonn

Number of standard Young tableaux of n cells and <= 9 columns.

Also the number of n-length words w over 9-ary alphabet {a1,a2,...,a9} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a9), where #(z,x) counts the letters x in word z.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner, From Dyck paths to standard Young tableaux, arXiv:1708.00513 [math.CO], 2017.

Column k=9 of 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, h([l[], 1$n]), `if`(i<1, 0,
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
a:= n-> g(n, 9, []):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember;
      `if`(n<5, [1, 1, 2, 4, 10][n+1],
      ((5*n^4+230*n^3+3574*n^2+20663*n+29393)*a(n-1)
       +7*(n-1)*(10*n^3+266*n^2+1919*n+2713)*a(n-2)
       -(n-1)*(n-2)*(230*n^2+3934*n+13587)*a(n-3)
       -3*(n-1)*(n-2)*(n-3)*(263*n+1414)*a(n-4)
       +945*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)) /
       ((n+20)*(n+8)*(n+18)*(n+14)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

Flatten[{1,RecurrenceTable[{-945 (-4+n) (-3+n) (-2+n) (-1+n) a[-5+n]+3 (-3+n) (-2+n) (-1+n) (1414+263 n) a[-4+n]+(-2+n) (-1+n) (13587+3934 n+230 n^2) a[-3+n]-7 (-1+n) (2713+1919 n+266 n^2+10 n^3) a[-2+n]+(-29393-20663 n-3574 n^2-230 n^3-5 n^4) a[-1+n]+(8+n) (14+n) (18+n) (20+n) a[n]==0,a[1]==1,a[2]==2,a[3]==4,a[4]==10,a[5]==26}, a, {n, 20}]}] (* Vaclav Kotesovec, Sep 11 2013 *)

a(n) ~ 14175/256 * 9^(n+18)/(Pi^2*n^18). - Vaclav Kotesovec, Sep 11 2013

A212916 Number of standard Young tableaux of n cells and height <= 10.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35695, 140140, 568360, 2389192, 10338315, 46118592, 211120144, 992316928, 4773362476, 23500234512, 118125854560, 606106812640, 3168660576795, 16872323635132, 91369920670420, 503022250919640, 2811920834508705

Offset: 0

Alois P. Heinz, May 30 2012

nonn

Number of standard Young tableaux of n cells and <= 10 columns.
Also the number of n-length words w over 10-ary alphabet {a1,a2,...,a10} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a10), where #(z,x) counts the letters x in word z.
Conjecture: generally (for tableaux with height <= k), a(n) ~ k^n/Pi^(k/2) * (k/n)^(k*(k-1)/4) * prod(j=1..k,Gamma(j/2)); set k=10 for this sequence. - Vaclav Kotesovec, Sep 12 2013

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=10 of 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, h([l[], 1$n]), `if`(i<1, 0,
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
a:= n-> g(n, 10, []):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember;
      `if`(n<6, [1, 1, 2, 4, 10, 26][n+1],
      ((70*n^4+4144*n^3+84986*n^2+685800*n+1656000)*a(n-1)
       +4*(n-1)*(35*n^4+1778*n^3+30106*n^2+184221*n+244350)*a(n-2)
       -8*(n-1)*(n-2)*(518*n^2+11916*n+59265)*a(n-3)
       -16*(n-1)*(n-2)*(n-3)*(259*n^2+4819*n+17355)*a(n-4)
       +21600*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)
       +14400*(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)) /
       ((n+21)*(n+9)*(n+16)*(n+25)*(n+24)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

Flatten[{1,RecurrenceTable[{-14400 (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) a[-6+n]-21600 (-4+n) (-3+n) (-2+n) (-1+n) a[-5+n]+16 (-3+n) (-2+n) (-1+n) (17355+4819 n+259 n^2) a[-4+n]+8 (-2+n) (-1+n) (59265+11916 n+518 n^2) a[-3+n]-4 (-1+n) (244350+184221 n+30106 n^2+1778 n^3+35 n^4) a[-2+n]-2 (828000+342900 n+42493 n^2+2072 n^3+35 n^4) a[-1+n]+(9+n) (16+n) (21+n) (24+n) (25+n) a[n]==0, a[1]==1, a[2]==2, a[3]==4, a[4]==10, a[5]==26, a[6]==76}, a, {n, 20}]}] (* Vaclav Kotesovec, Sep 11 2013 *)

a(n) ~ 42525/32 * 10^(n+45/2)/(Pi^(5/2)*n^(45/2)). - Vaclav Kotesovec, Sep 11 2013

cp's OEIS Frontend

A293109 Number T(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A293113 Number T(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet containing the k-th letter such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A293108 Number A(n,k) of multisets of nonempty words with a total of n letters over k-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A293112 Number A(n,k) of sets of nonempty words with a total of n letters over k-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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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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A240608 Number A(n,k) of n-length words w over a k-ary alphabet such that w is empty or a prefix z concatenated with letter a_i and i=1 or 0 < #(z,a_{i-1}) >= #(z,a_i), where #(z,a_i) counts the occurrences of the i-th letter in z; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A293110 Number of multisets of nonempty words with a total of n letters over n-ary alphabet such that within each prefix of a word every letter of the alphabet is at least as frequent as the subsequent alphabet letter.

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A007580 Number of Young tableaux of height <= 8.

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