A264051 Triangle read by rows: T(n,k) (n>=0, 0<=k<=A264078(n)) is the number of integer partitions of n having k standard Young tableaux such that no entries i and i+1 appear in the same row.

0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 2, 3, 0, 2, 4, 2, 1, 1, 1, 1, 1, 4, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 0, 0, 0, 1, 7, 2, 0, 0, 1, 0, 3, 0, 1, 0, 2, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 7, 3, 1, 2, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1

Offset

0,8

Comments

Row sums give A000041.

Column k=0 gives A025065(n-2) for n>=2.

Links

Alois P. Heinz, Rows n = 0..14, flattened

S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.

FindStat - Combinatorial Statistic Finder, Number of standard Young tableaux of an integer partition such that no k and k+1 appear in the same row.

Formula

Sum_{k=1..A264078(n)} k*T(n,k) = A237770(n). - Alois P. Heinz, Nov 02 2015

Example

Triangle begins:
0,1,
0,1,
1,1,
1,2,
2,2,1,
2,3,0,2,
4,2,1,1,1,1,1,
4,3,1,0,0,2,2,0,1,0,1,0,0,0,1,
7,2,0,0,1,0,3,0,1,0,2,1,0,0,0,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0,1,
...

Maple

h:= proc(l, j) option remember; `if`(l=[], 1,
      `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
      `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
       h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, x^h([1$n, l[]], 0),
      `if`(i<1, 0, g(n, i-1, l)+ `if`(i>n, 0,
       g(n-i, i, [i, l[]]))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(g(n$2, [])):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 02 2015

Mathematica

h[l_, j_] := h[l, j] = If[l == {}, 1, If[l[[1]] == 0, h[ReplacePart[l, 1 -> Sequence[]], j - 1], Sum[If[i != j && l[[i]] > 0 && (i == 1 || l[[i]] > l[[i - 1]]), h[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, Length[l]} ]]]; g[n_, i_, l_] := If[n == 0 || i == 1, x^h[Join[Array[1 &, n], l], 0], If[i < 1, 0, g[n, i - 1, l] + If[i > n, 0, g[n - i, i, Join[{i}, l]]] ]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][g[n, n, {}]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A001181, A237770, A264078.

Sequence in context: A373688 A192895 A210685 * A120965 A151931 A185636

Adjacent sequences: A264048 A264049 A264050 * A264052 A264053 A264054

Keyword

nonn,tabf

Author

Christian Stump, Nov 01 2015

Status

approved