

A046802


T(n, k) = Sum_{j=k..n} binomial(n, j)*E1(j, jk), where E1 are the Eulerian numbers A173018. Triangle read by rows, T(n, k) for 0 <= k <= n.


24



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, 63, 1, 1, 127, 1611, 5111, 5111, 1611, 127, 1, 1, 255, 5281, 26799, 44929, 26799, 5281, 255, 1, 1, 511, 16867, 131275, 344551, 344551, 131275, 16867, 511, 1, 1, 1023, 52905
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OFFSET

0,5


COMMENTS

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 hvectors of the stellahedra.  Tom Copeland, Oct 10 2014
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
The reciprocal of this entry's e.g.f. is [t e^(xt)  e^(x)] / (t1) = 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 [1t^(n+1)] / (1t) 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.)
The associated o.g.f. for the umbral inverses is Og(x) = x / (1x 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)xsqrt[12(1+t)x+((t1)x)^2]] / (2xt). Note that Og(x) gives the signed hpolynomials of the simplices and that Oginv(x) gives the hpolynomials 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 fpolynomials. (End)
The Appell polynomials p_n(x;t) in the formulas below specialize to the Swissknife 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
The row polynomials are the hpolynomials associated to the stellahedra, whose fpolynomials are the row polynomials of A248727. Cf. page 60 of the Buchstaber and Panov link.  Tom Copeland, Nov 08 2016
The row polynomials are the hpolynomials of the stellohedra, which enumerate partial permutations according to descents. Cf. Section 10.4 of the PostnikovReinerWilliams reference.  Lauren Williams, Jul 05 2022
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 (n1)simplices, respectively.  Tom Copeland, Jan 09 2017


REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, Holland, 1974, page 245 [From Roger L. Bagula, Nov 21 2009]
D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 6670.


LINKS

V. Buchstaber and T. Panov Toric Topology, arXiv:1210.2368v3 [math.AT], 2014.


FORMULA

p(t,x) = (1  x)*exp(t)/(1  x*exp(t*(1  x))).  Roger L. Bagula, Nov 21 2009
With offset=0, T(n,0)=1 otherwise T(n,k) = sum_{i=0..k1} C(n,i)((ik)^i*(ki+1)^(ni)  (ik+1)^i*(ki)^(ni)) (cf. Williams).  Tom Copeland, Oct 10 2014
A raising operator (with D = d/dx) associated with this entry's row polynomials is R = x + t + (1t) / [1t e^{(1t)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
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
From the previous umbral statement, OP(x,d/dy) y^n = (y + q.(x))^n, where OP(x,y) = exp[y * q.(x)] = (1x)/(1x*exp((1x)y)), the e.g.f. of A123125, so OP(x,d/dy) y^n evaluated at y = 1 is r_n(x), the nth row polynomial of this entry, with offset 0.  Tom Copeland, Jun 25 2018
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 hpolynomials 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 Swissknife 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 fpolynomials of the permutohedra, cf. also A019538) and A028246 (another version of the Worpitsky triangle).  Tom Copeland, Jan 24 2020
Sum_{k=0..n} (1)^k*T(n, k) = A122045(n).
Sum_{k=0..n} 2^(nk)*T(n,k) = A007047(n).
Sum_{k=0..n} T(n, nk) = A000522(n).
Sum_{k=0..n} T(nk, k) = Sum_{k=0..n} (n  k)^k = A026898(n1) for n >= 1.
Sum_{k=0..n} k*T(n, k) = A036919(n) = floor(n*n!*e/2).
(End)


EXAMPLE

The triangle T(n, k) begins:
n\k 0 1 2 3 4 5 6 7
0: 1
1: 1 1
2: 1 3 1
3: 1 7 7 1
4: 1 15 33 15 1
5: 1 31 131 131 31 1
6: 1 63 473 883 473 63 1
7: 1 127 1611 5111 5111 1611 127 1


MAPLE

T := (n, k) > add(binomial(n, r)*combinat:eulerian1(r, rk), r = k .. n):
for n from 0 to 8 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jun 27 2018


MATHEMATICA

t[_, 1] = 1; t[n_, n_] = 1; t[n_, 2] = 2^(n1)1;
t[n_, k_] = Sum[((ik+1)^i*(ki)^(ni1)  (ik+2)^i*(ki1)^(ni1))*Binomial[n1, i], {i, 0, k1}];
T[n_, k_] := t[n+1, k+1]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten
T[ n_, k_] := Coefficient[n! SeriesCoefficient[(1x) Exp[t] / (1  x Exp[(1x) t]), {t, 0, n}] // Simplify, x, k];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Michael Somos, Jan 22 2015 *)


CROSSREFS

Cf. A008292, A123125, A248727, A074909, A007318, A000225, A066810, A028246, A001263, A119879, A001586, A019538, A090582, A123125, A130850.


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



