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A238129 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having largest ascent k, n>=0, 0<=k<=n. 12
1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 8, 1, 0, 0, 1, 19, 5, 1, 0, 0, 1, 47, 21, 6, 1, 0, 0, 1, 114, 78, 31, 7, 1, 0, 0, 1, 286, 292, 133, 43, 8, 1, 0, 0, 1, 723, 1028, 586, 215, 57, 9, 1, 0, 0, 1, 1869, 3691, 2453, 1073, 325, 73, 10, 1, 0, 0, 1, 4870, 13004, 10357, 5058, 1836, 467, 91, 11, 1, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Also number of standard Young tableaux with a pair of cells (v,v+1) such that v lies k rows below v+1, and no pair (u,u+1) with a larger such separation exists.

LINKS

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..35, flattened

EXAMPLE

Triangle starts:

00: 1;

01: 1,     0;

02: 1,     1,     0;

03: 1,     3,     0,     0;

04: 1,     8,     1,     0,     0;

05: 1,    19,     5,     1,     0,    0;

06: 1,    47,    21,     6,     1,    0,    0;

07: 1,   114,    78,    31,     7,    1,    0,   0;

08: 1,   286,   292,   133,    43,    8,    1,   0,   0;

09: 1,   723,  1028,   586,   215,   57,    9,   1,   0,  0;

10: 1,  1869,  3691,  2453,  1073,  325,   73,  10,   1,  0, 0;

11: 1,  4870, 13004, 10357,  5058, 1836,  467,  91,  11,  1, 0, 0;

12: 1, 12943, 46452, 43462, 23953, 9631, 2941, 645, 111, 12, 1, 0, 0;

...

MAPLE

b:= proc(n, v, l) option remember; local m; m:=nops(l);

      `if`(n<1, 1, expand(add(`if`(i=1 or l[i-1]>l[i],

       (p->`if`(v<i, add(coeff(p, x, h)*`if`(h<i-v,

       x^(i-v), x^h), h=0..max(i-v, degree(p))), p))

       (b(n-1, i, subsop(i=l[i]+1, l))), 0), i=1..m)+

       (p->add(coeff(p, x, h)*`if`(h<m+1-v, x^(m+1-v), x^h),

       h=0..max(m+1-v, degree(p))))(b(n-1, m+1, [l[], 1]))))

    end:

T:= n->(p->seq(coeff(p, x, i), i=0..n))(b(n-1, 1, [1])):

seq(T(n), n=0..14);

MATHEMATICA

b[n_, v_, l_List] := b[n, v, l] = Module[{m = Length[l]}, If[n<1, 1, Expand[Sum[If[i == 1 || l[[i-1]]>l[[i]], Function[{p}, If[v<i, Sum[Coefficient[p, x, h]*If[h<i-v, x^(i-v), x^h], {h, 0, Max[i-v, Exponent[p, x]]}], p]][b[n-1, i, ReplacePart[l, i -> l[[i]]+1]]], 0], {i, 1, m}] + Function[{p}, Sum[Coefficient[p, x, h]*If[h<m+1-v, x^(m+1-v), x^h], {h, 0, Max[m+1-v, Exponent[p, x]]}]][b[n-1, m+1, Append[l, 1]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, 1, {1}]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 07 2015, translated from Maple *)

CROSSREFS

Columns k=0-10 give: A000012, A244208, A244198, A244199, A244200, A244201, A244202, A244203, A244204, A244205, A244206.

Row sums are A000085.

Cf. A238128.

Sequence in context: A144299 A060514 A176788 * A212221 A193291 A096936

Adjacent sequences:  A238126 A238127 A238128 * A238130 A238131 A238132

KEYWORD

nonn,tabl

AUTHOR

Joerg Arndt and Alois P. Heinz, Feb 21 2014

STATUS

approved

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Last modified June 26 13:27 EDT 2017. Contains 288766 sequences.