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A046802(x,y) --> A046802(x,y+1), transform of e.g.f. for the graded number of positroids of the totally nonnegative Grassmannians G+(k,n); enumerates faces of the stellahedra.
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%I #133 May 04 2024 14:55:31

%S 1,2,1,5,5,1,16,24,10,1,65,130,84,19,1,326,815,720,265,36,1,1957,5871,

%T 6605,3425,803,69,1,13700,47950,65646,44240,15106,2394,134,1,109601,

%U 438404,707840,589106,267134,63896,7094,263,1

%N A046802(x,y) --> A046802(x,y+1), transform of e.g.f. for the graded number of positroids of the totally nonnegative Grassmannians G+(k,n); enumerates faces of the stellahedra.

%C This is a transform of A046802 treating it as an array of h-vectors, so y is replaced by (y+1) in the e.g.f. for A046802.

%C An e.g.f. for the reversed row polynomials with signs is given by exp(a.(0;t)x) = [e^{(1+t)x} [1+t(1-e^(-x))]]^(-1) = 1 - (1+2t)x + (1+5t+5t^2)x^2/2! + ... . The reciprocal is an e.g.f. for the reversed face polynomials of the simplices A074909, i.e., exp(b.(0;t)x) = e^{(1+t)x} [1+t(1-e^(-x))] = 1 + (1+2t)x +(1+3t+3t^2) x^2/2! + ... , so the relations of A133314 apply between the two sets of polynomials. In particular, umbrally [a.(0;t)+b.(0;t)]^n vanishes except for n=0 for which it's unity, implying the two sets of Appell polynomials formed from the two bases, a_n(z;t) = (a.(0;t)+z)^n and b_n(z;t) = (b.(0;t) + z)^n, are an umbral compositional inverse pair, i.e., b_n(a.(x;t);t)= x^n = a_n(b.(x;t);t). Raising operators for these Appell polynomials are related to the polynomials of A028246, whose reverse polynomials are given by A123125 * A007318. Compare: A248727 = A007318 * A123125 * A007318 and A046802 = A007318 * A123125. See A074909 for definitions and related links. - _Tom Copeland_, Jan 21 2015

%C The o.g.f. for the umbral inverses is Og(x) = x / (1 - x b.(0;t)) = x / [(1-tx)(1-(1+t)x)] = x + (1+2t) x^2 + (1+3t+3t^2) x^3 + ... . Its compositional inverse is an o.g.f for signed A033282, the reverse f-polynomials for the simplicial duals of the Stasheff polytopes, or associahedra of type A, Oginv(x) =[1+(1+2t)x-sqrt[1+2(1+2t)x+x^2]] / (2t(1+t)x) = x - (1+2t) x^2 + (1+5t+5t^2) x^3 + ... . Contrast this with the o.g.f.s related to the corresponding h-polynomials in A046802. - _Tom Copeland_, Jan 24 2015

%C Face vectors, or coefficients of the face polynomials, of the stellahedra, or stellohedra. See p. 59 of Buchstaber and Panov. - _Tom Copeland_, Nov 08 2016

%C See A008279 for a relation between the e.g.f.s enumerating the faces of permutahedra and stellahedra. - _Tom Copeland_, Nov 14 2016

%H P. Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.

%H L. Berry, S. Forcey, M. Ronco, and P. Showers, <a href="https://arxiv.org/abs/1608.08546">Polytopes and Hopf algebras of painted trees: Fan graphs and Stellohedra</a>, arXiv:1608.08546 [math.CO], 2018.

%H L. Berry, S. Forcey, M. Ronco, and P. Showers, <a href="https://arxiv.org/abs/1608.08546">Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes</a>, arXiv:1608.08546 [math.CO], 2019.

%H V. Buchstaber and T. Panov, <a href="https://arxiv.org/abs/1210.2368">Toric Topology</a>, arXiv:1210.2368v3 [math.AT], 2014.

%H R. Da Rosa, D. Jensen, and D. Ranganathan, <a href="https://arxiv.org/abs/1411.0537">Toric graph associahedra and compactifications of M_(0,n)</a>, arXiv:1411.0537 [math.AG], 2015.

%H S. Forcey, M. Ronco, and P. Showers, <a href="http://arxiv.org/abs/1608.08546">Polytopes and algebras of grafted trees: Stellohedra</a>, arXiv:1608.08546v2 [math.CO], 2016.

%H Stefan Forcey, <a href="http://www.math.uakron.edu/~sf34/hedra.htm#index">The Hedra Zoo</a>

%H I. Limonchenko, <a href="https://arxiv.org/abs/1510.07778">Moment-angle manifolds, 2-truncated cubes and Massey operations</a>, arXiv:1510.07778 [math.AT], 2017.

%H M. Lin, <a href="https://www.math.hmc.edu/~mlin/thesis/mlin-2016-thesis.pdf">Graph Cohomology</a>, 2016, (Fig. 2.5 is a stellahedron).

%H T. Manneville and V. Pilaud, <a href="http://arxiv.org/abs/1501.07152">Compatibility fans for graphical nested complexes</a>, arXiv:1501.07152v3 [math.CO], 2015-2016.

%H MathOverflow, <a href="https://mathoverflow.net/questions/270070/analogue-of-conic-sections-for-the-permutohedra-associahedra-and-noncrossing-p">Analogue of conic sections for the permutohedra, associahedra, and noncrossing partitions</a>, an MO question posed by T. Copeland, 2017. (See Buchstaber references therein.)

%H V. Pilaud, <a href="https://www.mat.univie.ac.at/~slc/wpapers/s76vortrag/pilaud.pdf">The Associahedron and its Friends</a>, presentation for Séminaire Lotharingien de Combinatoire, April 4-6, 2016. [From _Tom Copeland_, Jun 26 2018]

%F Let M(n,k)= sum{i=0,..,k-1, C(n,i)[(i-k)^i*(k-i+1)^(n-i)- (i-k+1)^i*(k-i)^(n-i)]} with M(n,0)=1. Then M(n,k)= A046802(n,k), and T(n,j)= sum(k=j,..,n, C(k,j)*M(n,k)) for j>0 with T(n,0)= 1 + sum(k=1,..,n, M(n,k)) for n>0 and T(0,0)=1.

%F E.g.f: y * exp[x*(y+1)]/[y+1-exp(x*y)].

%F Row sums are A007047. Row polynomials evaluated at -1 are unity. Row polynomials evaluated at -2 are A122045.

%F First column is A000522. Second column appears to be A036918/2 = (A001339-1)/2 = n*A000522(n)/2.

%F Second diagonal is A052944. (Changed from conjecture to fact on Nov 08 2016.)

%F The raising operator for the reverse row polynomials with row signs is R = x - (1+t) - t e^(-D) / [1 + t(1-e^(-D))] evaluated at x = 0, with D = d/dx. Also R = x - d/dD log[exp(a.(0;t)D], or R = - d/dz log[e^(-xz) exp(a.(0;t)z)] = - d/dz log[exp(a.(-x;t)z)] with the e.g.f. defined in the comments and z replaced by D. Note that t e ^(-D) / [1+t(1-e^(-D))] = t - (t+t^2) D + (t+3t^2+2t^3) D^2/2! - ... is an e.g.f. for the signed reverse row polynomials of A028246. - _Tom Copeland_, Jan 23 2015

%F Equals A007318*(padded A090582)*A007318*A097808 = A007318*(padded (A008292*A007318))*A007318*A097808 = A007318*A130850 = A007318*(mirror of A028246). Padded means in the same way that A097805 is padded A007318. - _Tom Copeland_, Nov 14 2016

%F Umbrally, the row polynomials are p_n(x) = (1 + q.(x))^n, where (q.(x))^k = q_k(x) are the row polynomials of A130850. - _Tom Copeland_, Nov 16 2016

%F From the previous umbral statement, OP(x,d/dy) y^n = (y + q.(x))^n, where OP(x,y) = exp[y * q.(x)] = x/((1+x)*exp(-x*y) - 1), the e.g.f. of A130850, so OP(x,d/dy) y^n evaluated at y = 1 is p_n(x), the n-th row polynomial of this entry, with offset 0. - _Tom Copeland_, Jun 25 2018

%F Consolidating some formulas in this entry and A046082, in umbral notation for concision, with all offsets 0: Let A_n(x;y) = (y + E.(x))^n, an Appell sequence in y where E.(x)^k = E_k(x) are the Eulerian polynomials of A123125. Then the row polynomials of A046802 (the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of this entry (A248727, the face polynomials of the stellahedra), by f_n(x) = A_n(1 + x;1); the Swiss-knife polynomials of A119879, by Sw_n(x) = A_n(-1;1 + x); and the row polynomials of the Worpitsky triangle (A130850), by w_n(x) = A(1 + x;0). Other specializations of A_n(x;y) give A090582 (the f-polynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle). - _Tom Copeland_, Jan 24 2020

%e The triangle T(n, k) starts:

%e n\k 0 1 2 3 4 5 6 7 ...

%e 1: 1

%e 2: 2 1

%e 3: 5 5 1

%e 4: 16 24 10 1

%e 5: 65 130 84 19 1

%e 6: 326 815 720 265 36 1

%e 7: 1957 5871 6605 3425 803 69 1

%e 8: 13700 47950 65646 44240 15106 2394 134 1

%e ... reformatted, _Wolfdieter Lang_, Mar 27 2015

%t (* t = A046802 *) t[_, 1] = 1; t[n_, n_] = 1; t[n_, 2] = 2^(n - 1) - 1; t[n_, k_] = Sum[((i - k + 1)^i*(k - i)^(n - i - 1) - (i - k + 2)^i*(k - i - 1)^(n - i - 1))*Binomial[n - 1, i], {i, 0, k - 1}]; T[n_, j_] := Sum[Binomial[k, j]*t[n + 1, k + 1], {k, j, n}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jan 23 2015, after _Tom Copeland_ *)

%Y Cf. A046802, A007047, A122045, A000522, A036918, A001339, A052944.

%Y Cf. A074909, A028246, A133314, A007318, A123125, A033282.

%Y Cf. A008279, A008292, A090582, A097808, A130850.

%Y Cf. A019538, A119879.

%K easy,nonn,tabl

%O 0,2

%A _Tom Copeland_, Oct 12 2014

%E Title expanded by _Tom Copeland_, Nov 08 2016