A238728 -id:A238728 - cp's OEIS frontend

A238727 Number T(n,k) of standard Young tableaux with n cells where k is the largest value in the last row; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 1, 2, 7, 0, 0, 1, 3, 8, 14, 0, 0, 1, 4, 11, 19, 41, 0, 0, 1, 7, 19, 34, 64, 107, 0, 0, 1, 11, 32, 62, 119, 202, 337, 0, 0, 1, 21, 64, 131, 248, 418, 671, 1066, 0, 0, 1, 36, 124, 277, 545, 943, 1518, 2361, 3691

Offset: 0

Joerg Arndt and Alois P. Heinz, Mar 03 2014

T(0,0) = 1 by convention.
Also the number of ballot sequences of length n having the last occurrence of the maximal value at position k.
T(n,3) = A051920(n-3) for n>3.
T(2*n,n) gives A246818.
Main diagonal gives A238728.
Row sums give A000085.

			The 10 tableaux with n=4 cells sorted by largest value in the last row:
  :[1 3 4]:[1 4] [1 2 4]:[1] [1 2] [1 3] [1 2 3] [1 2] [1 3] [1 2 3 4]:
  :[2]    :[2]   [3]    :[2] [3]   [2]   [4]     [3 4] [2 4]          :
  :       :[3]          :[3] [4]   [4]                                :
  :       :             :[4]                                          :
  : --2-- : -----3----- : ---------------------4--------------------- :
The 10 ballot sequences of length 4 sorted by the position of the last occurrence of the maximal value:
  [1, 2, 1, 1]  ->  2 } -- 1
  [1, 2, 3, 1]  ->  3 \ __ 2
  [1, 1, 2, 1]  ->  3 /
  [1, 2, 3, 4]  ->  4 \
  [1, 1, 2, 3]  ->  4  \
  [1, 2, 1, 3]  ->  4   \
  [1, 1, 1, 2]  ->  4    } 7
  [1, 1, 2, 2]  ->  4   /
  [1, 2, 1, 2]  ->  4  /
  [1, 1, 1, 1]  ->  4 /
thus row 4 = [0, 0, 1, 2, 7].
Triangle T(n,k) begins:
  00:   1;
  01:   0, 1;
  02:   0, 0, 2;
  03:   0, 0, 1,  3;
  04:   0, 0, 1,  2,   7;
  05:   0, 0, 1,  3,   8,  14;
  06:   0, 0, 1,  4,  11,  19,  41;
  07:   0, 0, 1,  7,  19,  34,  64, 107;
  08:   0, 0, 1, 11,  32,  62, 119, 202,  337;
  09:   0, 0, 1, 21,  64, 131, 248, 418,  671, 1066;
  10:   0, 0, 1, 36, 124, 277, 545, 943, 1518, 2361, 3691;

Joerg Arndt and Alois P. Heinz, Rows n = 0..43, flattened
Wikipedia, Young tableau

h:= proc(l) option remember; local n, s; n:= nops(l); s:= add(i, i=l);
     `if`(n=0, 1, add(`if`(il[i+1], h(subsop(i=l[i]-1, l)),
     `if`(i=n, (p->add(coeff(p,x,j)*x^`if`(j1, l[i]-1, [][]), l))), 0)), i=1..n))
    end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]),
       add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i)):
T:= n-> (p->seq(coeff(p, x, i), i=0..n))(g(n$2, [])):
seq(T(n), n=0..12);

h[l_] := h[l] = With[{n = Length[l], s = Total[l]},
     If[n == 0, 1, Sum[If[i < n && l[[i]] > l[[i + 1]],
     h[ReplacePart[l, i -> l[[i]] - 1]], If[i == n, Function[p,
     Sum[Coefficient[p, x, j] x^If[j < s, s, j], {j, 0,
     Exponent[p, x]}]][h[ReplacePart[l, i -> If[l[[i]] > 1,
     l[[i]] - 1, Nothing]]]], 0]], {i, n}]]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Table[1, {n}]]],
     Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
T[n_] := CoefficientList[g[n, n, {}], x];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Aug 27 2021, after Maple code *)

A369589 Triangular array read by rows: T(m,n) = number of Yamanouchi words over the alphabet {1, 2, ..., n} of length m that start with n, m >= 1, n = 1..m.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 4, 5, 3, 1, 1, 10, 14, 11, 4, 1, 1, 15, 35, 31, 19, 5, 1, 1, 35, 90, 106, 69, 29, 6, 1, 1, 56, 245, 323, 265, 127, 41, 7, 1, 1, 126, 644, 1093, 971, 583, 209, 55, 8, 1, 1, 210, 1716, 3439, 3644, 2446, 1106, 319, 71, 9, 1, 1, 462, 4707, 11716, 13771, 10616, 5323, 1904, 461, 89, 10, 1

Offset: 1

Max Alekseyev, Jan 26 2024

			Array starts with
  m=1: 1
  m=2: 1,  1
  m=3: 1,  1,  1
  m=4: 1,  3,  2,  1
  m=5: 1,  4,  5,  3,  1
  m=6: 1, 10, 14, 11,  4, 1
  m=7: 1, 15, 35, 31, 19, 5, 1

Wikipedia, Lattice word.

Row sums: A238728.
Columns: A000012 (n=1), A037952 (n=2), A369591 (n=3).
Cf. A001006, A369588.

A369591 a(n) = number of Yamanouchi words over the alphabet {1, 2, 3} of length n that start with 3.

Original entry on oeis.org

0, 0, 1, 2, 5, 14, 35, 90, 245, 644, 1716, 4707, 12727, 34683, 96174, 264784, 732785, 2051338, 5719684, 16007081, 45160093, 127106181, 358743903, 1018566177, 2887666445, 8204375034

Offset: 1

Max Alekseyev, Jan 26 2024

Column 3 in A369589.
Number of Yamanouchi words over the alphabet {1, 2, 3} is given by A001006, and by A217323 for the words containing 3.
Cf. A238728, A369590.

cp's OEIS Frontend

A238727 Number T(n,k) of standard Young tableaux with n cells where k is the largest value in the last row; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A369589 Triangular array read by rows: T(m,n) = number of Yamanouchi words over the alphabet {1, 2, ..., n} of length m that start with n, m >= 1, n = 1..m.

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A369591 a(n) = number of Yamanouchi words over the alphabet {1, 2, 3} of length n that start with 3.

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