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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Jan 22 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A001181, A237770, A264078.

Sequence in context: A147680 A192895 A210685 * A120965 A151931 A185636

Adjacent sequences:  A264048 A264049 A264050 * A264052 A264053 A264054

KEYWORD

nonn,tabf

AUTHOR

Christian Stump, Nov 01 2015

STATUS

approved

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Last modified August 21 04:54 EDT 2019. Contains 326162 sequences. (Running on oeis4.)