The OEIS is supported by the many generous donors to the OEIS Foundation.

 Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”). Other ways to Give
 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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(2n,n) gives A246818. Main diagonal gives A238728. Row sums give A000085. LINKS Joerg Arndt and Alois P. Heinz, Rows n = 0..43, flattened Wikipedia, Young tableau EXAMPLE 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; MAPLE 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); MATHEMATICA 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, 12}] // Flatten (* Jean-François Alcover, Aug 27 2021, after Maple code )* CROSSREFS Sequence in context: A116489 A166373 A202451 * A056885 A029373 A357645 Adjacent sequences: A238724 A238725 A238726 * A238728 A238729 A238730 KEYWORD nonn,tabl AUTHOR Joerg Arndt and Alois P. Heinz, Mar 03 2014 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 4 16:41 EST 2022. Contains 358563 sequences. (Running on oeis4.)