Tricia Muldoon Brown - cp's OEIS frontend

User: Tricia Muldoon Brown

Tricia Muldoon Brown has authored 4 sequences.

A320657 a(n) is the number of non-unimodal sequences with n nonzero terms that arise as a convolution of sequences of binomial coefficients preceded by a finite number of ones.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 7, 12, 16, 24, 30, 41, 50, 65, 77, 96, 112, 136, 156, 185, 210, 245, 275, 316, 352, 400, 442, 497, 546, 609, 665, 736, 800, 880, 952, 1041, 1122, 1221, 1311, 1420, 1520, 1640, 1750, 1881, 2002, 2145, 2277, 2432, 2576, 2744, 2900, 3081, 3250, 3445, 3627, 3836, 4032

Offset: 1

Tricia Muldoon Brown, Oct 17 2018

nonn

For integers x,y,p,q >= 0, set (s_i){i>=1} to be the sequence of p ones followed by the binomial coefficients C(x,j) for 0 <= j <= x followed by an infinite string of zeros, and set (t_i){i>=1} to be the sequence of q ones followed by the binomial coefficients C(y,j) for 0 <= j <= y followed by an infinite string of zeros. Then a(n) is the number of non-unimodal sequences (r_i){i>=1} where r_i = Sum{j=1..i} s_j*t_{i-j} for some(s_i) and (t_i) such that x + y + p + q + 1 = n.

Let T be a rooted tree created by identifying the root vertices of two broom graphs. a(n) is the number of trees T on n vertices whose poset of connected, vertex-induced subgraphs is not rank unimodal.

T. M. Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018). See table 1 on page 17.
M. Jacobson, A. E. Kézdy, and S. Seif, The poset on connected induced subgraphs of a graph need not be Sperner, Order, 12 (1995) 315-318.

Cf. A005993, A024206. Equals A005581 for n even.

Table[If[EvenQ[n], 2*(Sum[Floor[i(i+4)/4], {i,0,(n/2)}]) - Floor[n^2/16], 2*(Sum[Floor[i(i+4)/4], {i,0,(n-1)/2}]) - Floor[(n-1)^2/16] + Floor[(n+1)(n+9)/16]], {n,0,40}]

a(n+10) = 2*(Sum_{i=1..n/2} floor(i*(i+4)/4)) - floor(n^2/16) for n even.

a(n+10) = 2*(Sum_{i=1..(n-1)/2} floor(i(i+4)/4)) - floor((n-1)^2/16) + floor((n+1)*(n+9)/16) for n odd.

A275662 Triangle read by rows: T(n,k) = number of convex domino towers with n dominoes having widest row with k dominoes.

Original entry on oeis.org

1, 3, 1, 7, 6, 1, 15, 18, 7, 1, 31, 48, 17, 9, 1, 63, 109, 49, 20, 11, 1, 127, 240, 115, 52, 24, 13, 1, 255, 498, 258, 122, 61, 28, 15, 1, 511, 1026, 551, 261, 136, 71, 32, 17, 1, 1023, 2065, 1163, 531, 298, 157, 81, 36, 19, 1

Offset: 1

Tricia Muldoon Brown, Aug 04 2016

A domino tower is built by placing dominoes horizontally on a convex horizontal base. A domino tower is convex if all its columns and rows are convex.

			Triangle begins:
1;
3, 1;
7, 6, 1;
15, 18, 7, 1;
...
If n = 3 and k = 2, the widest row of the domino tower has two dominoes. Thus the third domino may be found supporting the row of two dominoes in one way or being supported by the row of two dominoes in 5 ways, so T(3,2) = 6.

T. M. Brown, Convex domino towers, arXiv:1608.01562 [math.CO] (2016).

Column 1: A000225, n>=1.

Cf. A117468, A275204, A275599.

G.f.: (2*A_k(x)+B_k(x))*(C_{k-1}(x)+1) where A_k(x) is the generating function on right-skewed domino towers with a base of k dominoes from the sequence A275599, B_k(x) is the generating function on domino stacks with a base of k dominoes associated with the sequence A275204, and C_k(x) is the generating function on flat partitions whose largest part is k-1 given by the sequence A117468.

A275599 Triangle read by rows: T(n,k) = number of right-skewed domino towers with n dominoes having a base of k dominoes placed end-to-end.

Original entry on oeis.org

1, 3, 1, 7, 4, 1, 15, 12, 4, 1, 31, 27, 13, 4, 1, 63, 61, 34, 13, 4, 1, 127, 124, 77, 35, 13, 4, 1, 255, 258, 165, 86, 35, 13, 4, 1, 511, 513, 348, 185, 87, 35, 13, 4, 1, 1023, 1039, 698, 399, 196, 87, 35, 13, 4, 1, 2047, 2062, 1410, 811, 423, 197, 87, 35, 13, 4, 1

Offset: 2

Tricia Muldoon Brown, Aug 03 2016

Domino towers are created by stacking domino blocks horizontally on a convex base of k dominoes. A right-skewed domino tower is a parallelogram domino tower such that at least one column of the polyomino is to the right of the base.

			Triangle begins:
1;
3, 1;
7, 4, 1;
15, 12, 4, 1;
...
For n = 5 and k = 3, each tower has a convex base of three dominoes. The fourth domino may be placed directly above the rightmost domino of the base, in which case the fifth domino must be placed on the fourth domino so its right end is not above the base. Alternately, the fourth domino may be placed so its right end is not above the base, leaving three choices for the fifth domino: directly above, above and to the right, or directly to the left on the same level. Thus T(5,3) = 4.

T. M. Brown Convex domino towers, arXiv:1608.01562 [math.CO], (2016)
Tricia Muldoon Brown, Examples of right- or left-skewed domino towers of 10 dominoes having a base of 4 dominoes

Column 1: A000225, n>=1.

Cf. A275204.

T(n,k) = Sum_{i=1..k} 2*T(n-k,i)+A(n-k,i) where A(n,k) is given by A275204 and with initial conditions T(n+1,n)=1 and T(n,k)=0 if n<2 and k<1, or n

G.f.: x^k/(1-2x^k) Sum_{i=1..k}*A_k(x)*(Sum_{Subsets S of {i,i+1,..,k-1}} (Product_{j in S} 2x^j/(1-2x^k)) where A_k(x) is the generating function in A275204.

A275204 Triangle read by rows: T(n,k) = number of domino stacks with n dominoes having a base of k dominoes.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 5, 1, 1, 6, 8, 7, 1, 1, 7, 15, 12, 9, 1, 1, 9, 22, 25, 16, 11, 1, 1, 10, 31, 43, 35, 20, 13, 1, 1, 12, 41, 68, 65, 45, 24, 15, 1, 1, 13, 54, 99, 113, 87, 55, 28, 17, 1, 1, 15, 66, 143, 178, 159, 109, 65, 32, 19, 1, 1, 16, 82, 193, 273, 267, 205, 131, 75, 36, 21, 1, 1, 18, 98, 258, 398, 430, 357, 251, 153, 85, 40, 23, 1

Offset: 1

Tricia Muldoon Brown, Jul 19 2016

The k-th column is the number of domino stacks having a base of k dominoes.
The n-th row is the number of domino stacks consisting of n dominoes.
A domino stack corresponds to a convex polyomino built with dominoes such that all columns intersect the base.

			Triangle begins:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  5,  1;
  1,  6,  8,  7,  1;
  1,  7, 15, 12,  9,  1;
  ...
Dominoes are assumed to be horizontal, and each row must be a subset of the row below it. For n=4 and k=2, the bottom row has 2 dominoes. One possibility is to put both remaining dominoes in the next row up. Otherwise there will be one domino in the next row up, and it can be in three possible positions: right, center, or left. The last domino must be placed on top of it. So there are a total of four possible stacks, and therefore T(4,2) = 4. - _Michael B. Porter_, Jul 20 2016

Alois P. Heinz, Rows n = 1..200, flattened
T. M. Brown Convex Domino Towers, J. Integer Seq. 20 (2017).

T:= proc(n, k) option remember; `if`(k<1 or k>n, 0,
      `if`(n=k, 1, add((2*(k-i)+1)*T(n-k, i), i=1..k)))
    end:
seq(seq(T(n,k), k=1..n), n=1..15);  # Alois P. Heinz, Jul 19 2016

T[n_, n_] = 1; T[n_, k_] /; n<1 || k<1 || nJean-François Alcover, Aug 17 2018 *)

T(n,k) = Sum_{i=1..k} (2*(k-i)+1)*T(n-k,i) where T(n,n)=1 and T(n,k)=0 if n or k is nonpositive or if n is less than k.

G.f.: x^k/(1-x^k) Sum_{S} (Product_{i=1..j} (2*(k_{i+1}-k_i)+1)*x^(k_i)/ (1-x^(k_i))) where the sum is over all subsets S of {1,..,k-1} such that S={k_1

Data corrected by Jean-François Alcover, Aug 17 2018

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User: Tricia Muldoon Brown

A320657 a(n) is the number of non-unimodal sequences with n nonzero terms that arise as a convolution of sequences of binomial coefficients preceded by a finite number of ones.

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A275662 Triangle read by rows: T(n,k) = number of convex domino towers with n dominoes having widest row with k dominoes.

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A275599 Triangle read by rows: T(n,k) = number of right-skewed domino towers with n dominoes having a base of k dominoes placed end-to-end.

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A275204 Triangle read by rows: T(n,k) = number of domino stacks with n dominoes having a base of k dominoes.

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