A262169 -id:A262169

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A262163

Number A(n,k) of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

+10
12

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 2, 0, 1, 1, 2, 4, 5, 0, 1, 1, 2, 5, 16, 16, 0, 1, 1, 2, 5, 19, 54, 61, 0, 1, 1, 2, 5, 20, 82, 324, 272, 0, 1, 1, 2, 5, 20, 86, 454, 1532, 1385, 0, 1, 1, 2, 5, 20, 87, 516, 2795, 12256, 7936, 0, 1, 1, 2, 5, 20, 87, 521, 3135, 20346, 74512, 50521, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,13

LINKS

Alois P. Heinz, Antidiagonals n = 0..140, flattened

FORMULA

A(n,k) = Sum_{i=0..k} A258829(n,i).

EXAMPLE

Square array A(n,k) begins:

1, 1, 1, 1, 1, 1, 1, 1, ...

0, 1, 1, 1, 1, 1, 1, 1, ...

0, 1, 2, 2, 2, 2, 2, 2, ...

0, 2, 4, 5, 5, 5, 5, 5, ...

0, 5, 16, 19, 20, 20, 20, 20, ...

0, 16, 54, 82, 86, 87, 87, 87, ...

0, 61, 324, 454, 516, 521, 522, 522, ...

0, 272, 1532, 2795, 3135, 3264, 3270, 3271, ...

MAPLE

b:= proc(u, o, c) option remember; `if`(c<0, 0, `if`(u+o=0, x^c,

(p-> add(coeff(p, x, i)*x^max(i, c), i=0..degree(p)))(add(

b(u-j, o-1+j, c-1), j=1..u)+add(b(u+j-1, o-j, c+1), j=1..o))))

end:

A:= (n, k)-> (p-> add(coeff(p, x, i), i=0..min(n, k)))(b(0, n, 0)):

seq(seq(A(n, d-n), n=0..d), d=0..12);

MATHEMATICA

b[u_, o_, c_] := b[u, o, c] = If[c < 0, 0, If[u + o == 0, x^c, Function[p, Sum[Coefficient[p, x, i]*x^Max[i, c], {i, 0, Exponent[p, x]}]][Sum[b[u - j, o - 1 + j, c - 1], {j, 1, u}] + Sum[b[u + j - 1, o - j, c + 1], {j, 1, o}]]]]; A[n_, k_] := Function[p, Sum[Coefficient[p, x, i], {i, 0, Min[n, k]}]][b[0, n, 0]]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Feb 22 2016, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000007, A000111 for n>0, A262164, A262165, A262166, A262167, A262168, A262169, A262170, A262171, A262172.

Main diagonal gives: A258830.

Cf. A258829, A262124.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 13 2015

STATUS

approved

A316394

Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of seven.

+10
2

1, 7, 522, 4260, 163871, 1572713, 49601660, 554432537, 16431601190, 211104220038, 6214132488281, 90601727479330, 2718687446733807, 44477388811619142, 1378374571651666083, 25055072909382001747, 807272266530396465622, 16165637154045080226474

(list; graph; refs; listen; history; text; internal format)

OFFSET

7,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 7..451

FORMULA

a(n) = A262169(n) - A262168(n).

MAPLE

b:= proc(u, o, c, k) option remember;

`if`(c<0 or c>k, 0, `if`(u+o=0, 1,

add(b(u-j, o-1+j, c+1, k), j=1..u)+

add(b(u+j-1, o-j, c-1, k), j=1..o)))

end:

a:= n-> b(n, 0$2, 7)-b(n, 0$2, 6):

seq(a(n), n=7..24);

CROSSREFS

Column k=7 of A258829.

Cf. A262168, A262169.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jul 01 2018

STATUS

approved

A316395

Number of permutations p of [n] such that the up-down signature of 0,p has nonnegative partial sums with a maximal value of eight.

+10
2

1, 8, 1040, 9468, 507355, 5313447, 214378961, 2571977379, 92953037066, 1265907917962, 44038999833044, 674142774632948, 23379215615715958, 398561935596289153, 14037530250073013445, 264291741199540446059, 9551899031473405653870, 197148463934806397523934

(list; graph; refs; listen; history; text; internal format)

OFFSET

8,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 8..451

FORMULA

a(n) = A262170(n) - A262169(n).

MAPLE

b:= proc(u, o, c, k) option remember;

`if`(c<0 or c>k, 0, `if`(u+o=0, 1,

add(b(u-j, o-1+j, c+1, k), j=1..u)+

add(b(u+j-1, o-j, c-1, k), j=1..o)))

end:

a:= n-> b(n, 0$2, 8)-b(n, 0$2, 7):

seq(a(n), n=8..25);

CROSSREFS

Column k=8 of A258829.

Cf. A262169, A262170.

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jul 01 2018

STATUS

approved

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