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%I #182 Jun 16 2023 05:56:55
%S 1,1,1,1,3,1,1,7,7,1,1,15,33,15,1,1,31,131,131,31,1,1,63,473,883,473,
%T 63,1,1,127,1611,5111,5111,1611,127,1,1,255,5281,26799,44929,26799,
%U 5281,255,1,1,511,16867,131275,344551,344551,131275,16867,511,1,1,1023,52905
%N T(n, k) = Sum_{j=k..n} binomial(n, j)*E1(j, j-k), where E1 are the Eulerian numbers A173018. Triangle read by rows, T(n, k) for 0 <= k <= n.
%C T(n,k) is the number of positroid cells of the totally nonnegative Grassmannian G+(k,n) (cf. Postnikov/Williams). It is the triangle of the h-vectors of the stellahedra. - _Tom Copeland_, Oct 10 2014
%C See A248727 for a simple transformation of the row polynomials of this entry that produces the umbral compositional inverses of the polynomials of A074909, related to the face polynomials of the simplices. - _Tom Copeland_, Jan 21 2015
%C From _Tom Copeland_, Jan 24 2015: (Start)
%C The reciprocal of this entry's e.g.f. is [t e^(-xt) - e^(-x)] / (t-1) = 1 - (1+t) x + (1+t+t^2) x^2/2! - (1+t+t^2+t^3) x^3/3! + ... = e^(q.(0;t)x), giving the base sequence (q.(0;t))^n = q_n(0;t) = (-1)^n [1-t^(n+1)] / (1-t) for the umbral compositional inverses (q.(0;t)+z)^n = q_n(z;t) of the Appell polynomials associated with this entry, p_n(z;t) below, i.e., q_n(p.(z;t)) = z^n = p_n(q.(z;t)), in umbral notation. The relations in A133314 then apply between the two sets of base polynomials. (Inserted missing index in a formula - Mar 03 2016.)
%C The associated o.g.f. for the umbral inverses is Og(x) = x / (1-x q.(0:t)) = x / [(1+x)(1+tx)] = x / [1+(1+t)x+tx^2]. Applying A134264 to h(x) = x / Og(x) = 1 + (1+t) x + t x^2 leads to an o.g.f. for the Narayana polynomials A001263 as the comp. inverse Oginv(x) = [1-(1+t)x-sqrt[1-2(1+t)x+((t-1)x)^2]] / (2xt). Note that Og(x) gives the signed h-polynomials of the simplices and that Oginv(x) gives the h-polynomials of the simplicial duals of the Stasheff polynomials, or type A associahedra. Contrast this with A248727 = A046802 * A007318, which has o.g.f.s related to the corresponding f-polynomials. (End)
%C The Appell polynomials p_n(x;t) in the formulas below specialize to the Swiss-knife polynomials of A119879 for t = -1, so the Springer numbers A001586 are given by 2^n p_n(1/2;-1). - _Tom Copeland_, Oct 14 2015
%C The row polynomials are the h-polynomials associated to the stellahedra, whose f-polynomials are the row polynomials of A248727. Cf. page 60 of the Buchstaber and Panov link. - _Tom Copeland_, Nov 08 2016
%C The row polynomials are the h-polynomials of the stellohedra, which enumerate partial permutations according to descents. Cf. Section 10.4 of the Postnikov-Reiner-Williams reference. - _Lauren Williams_, Jul 05 2022
%C From p. 60 of the Buchstaber and Panov link, S = P * C / T where S, P, C, and T are the bivariate e.g.f.s of the h vectors of the stellahedra, permutahedra, hypercubes, and (n-1)-simplices, respectively. - _Tom Copeland_, Jan 09 2017
%D L. Comtet, Advanced Combinatorics, Reidel, Holland, 1974, page 245 [From _Roger L. Bagula_, Nov 21 2009]
%D D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.
%H Michael De Vlieger, <a href="/A046802/b046802.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Barry4/barry271.html">General Eulerian Polynomials as Moments Using Exponential Riordan Arrays</a>, Journal of Integer Sequences, 16 (2013), #13.9.6.
%H Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H Paul Barry, <a href="https://arxiv.org/abs/2107.14278">Series reversion with Jacobi and Thron continued fractions</a>, arXiv:2107.14278 [math.NT], 2021.
%H Carolina Benedetti, Anastasia Chavez, and Daniel Tamayo, <a href="https://arxiv.org/abs/1912.06873">Quotients of uniform positroids</a>, arXiv:1912.06873 [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 Jan Geuenich, <a href="https://arxiv.org/abs/1803.10707">Tilting modules for the Auslander algebra of K[x]/(xn)</a>, arXiv:1803.10707 [math.RT], 2018.
%H Jun Ma, Shi-mei Ma, Yeong-Nan Yeh, <a href="https://arxiv.org/abs/1711.09016">Recurrence relations for binomial-eulerian polynomials</a>, arXiv:1711.09016, Thm. 2.1.
%H A. Postnikov, <a href="http://arxiv.org/abs/math/0609764">Total positivity, Grassmannians, and networks</a>, arXiv:math/0609764 [math.CO], 2006.
%H A. Postnikov, V. Reiner, and L. Williams, <a href="https://arxiv.org/abs/math/0609184">Faces of generalized permutohedra</a>, arXiv:math/0609184 [math.CO], 2006-2007.
%H D. Singh, <a href="/A002627/a002627.pdf">The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers</a>, Math. Student, 20 (1952), 66-70. [Annotated scanned copy]
%H L. K. Williams, <a href="https://arxiv.org/abs/math/0307271">Enumeration of totally positive Grassmann cells</a>, arXiv:math/0307271 [math.CO], 2003-2004.
%H L. Williams, <a href="http://people.maths.ox.ac.uk/lmason/NGSA14/Slides/Lauren-Williams.pdf">The Positive Grassmannian (from a mathematician's perspective)</a>, 2014
%F E.g.f.: (y-1)*exp(x*y)/(y-exp((y-1)*x)). - _Vladeta Jovovic_, Sep 20 2003
%F p(t,x) = (1 - x)*exp(t)/(1 - x*exp(t*(1 - x))). - _Roger L. Bagula_, Nov 21 2009
%F With offset=0, T(n,0)=1 otherwise T(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)) (cf. Williams). - _Tom Copeland_, Oct 10 2014
%F With offset 0, T = A007318 * A123125. Second column is A000225; 3rd, appears to be A066810. - _Tom Copeland_, Jan 23 2015
%F A raising operator (with D = d/dx) associated with this entry's row polynomials is R = x + t + (1-t) / [1-t e^{(1-t)D}] = x + t + 1 + t D + (t+t^2) D^2/2! + (t+4t^2+t^3) D^3/3! + ... , containing the e.g.f. for the Eulerian polynomials of A123125. Then R^n 1 = (p.(0;t)+x)^n = p_n(x;t) are the Appell polynomials with this entry's row polynomials p_n(0;t) as the base sequence. Examples of this formalism are given in A028246 and A248727. - _Tom Copeland_, Jan 24 2015
%F With offset 0, T = A007318*(padded A090582)*(inverse of A097805) = A007318*(padded A090582)*(padded A130595) = A007318*A123125 = A007318*(padded A090582)*A007318*A097808*A130595, where padded matrices are of the form of padded A007318, which is A097805. Inverses of padded matrices are just the padded versions of inverses of the unpadded matrices. This relates the face vectors, or f-vectors, and h-vectors of the permutahedra / permutohedra to those of the stellahedra / stellohedra. - _Tom Copeland_, Nov 13 2016
%F Umbrally, the row polynomials (offset 0) are r_n(x) = (1 + q.(x))^n, where (q.(x))^k = q_k(x) are the row polynomials of A123125. - _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)] = (1-x)/(1-x*exp((1-x)y)), the e.g.f. of A123125, so OP(x,d/dy) y^n evaluated at y = 1 is r_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 A248727, 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 this entry (A046802, the h-polynomials of the stellahedra) are given by h_n(x) = A_n(x;1); the row polynomials of 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
%F From _Peter Luschny_, Apr 30 2021: (Start)
%F Sum_{k=0..n} (-1)^k*T(n, k) = A122045(n).
%F Sum_{k=0..n} 2^(n-k)*T(n,k) = A007047(n).
%F Sum_{k=0..n} T(n, n-k) = A000522(n).
%F Sum_{k=0..n} T(n-k, k) = Sum_{k=0..n} (n - k)^k = A026898(n-1) for n >= 1.
%F Sum_{k=0..n} k*T(n, k) = A036919(n) = floor(n*n!*e/2).
%F (End)
%e The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7
%e 0: 1
%e 1: 1 1
%e 2: 1 3 1
%e 3: 1 7 7 1
%e 4: 1 15 33 15 1
%e 5: 1 31 131 131 31 1
%e 6: 1 63 473 883 473 63 1
%e 7: 1 127 1611 5111 5111 1611 127 1
%e ... Reformatted. - _Wolfdieter Lang_, Feb 14 2015
%p T := (n, k) -> add(binomial(n, r)*combinat:-eulerian1(r, r-k), r = k .. n):
%p for n from 0 to 8 do seq(T(n, k), k=0..n) od; # _Peter Luschny_, Jun 27 2018
%t t[_, 1] = 1; t[n_, n_] = 1; t[n_, 2] = 2^(n-1)-1;
%t 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 T[n_, k_] := t[n+1, k+1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten
%t (* _Jean-François Alcover_, Jan 22 2015, after _Tom Copeland_ *)
%t T[ n_, k_] := Coefficient[n! SeriesCoefficient[(1-x) Exp[t] / (1 - x Exp[(1-x) t]), {t, 0, n}] // Simplify, x, k];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* _Michael Somos_, Jan 22 2015 *)
%Y Row sums give A000522.
%Y Cf. A008292, A123125, A248727, A074909, A007318, A000225, A066810, A028246, A001263, A119879, A001586, A019538, A090582, A123125, A130850.
%Y Cf. A122045, A007047, A000522, A026898, A036919.
%K nonn,tabl,easy,nice
%O 0,5
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Sep 20 2003
%E First formula corrected by _Wolfdieter Lang_, Feb 14 2015
%E Offset set to 0 and edited by _Peter Luschny_, Apr 30 2021
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